They also lost a great, big slice of humanity, with all its warts and wonderments.
In fact, they might have lost the last vestige of true humanity in their professional but increasingly cold and corporate clubhouse.
A-Rod may not have been the perfect man, the perfect teammate or the perfect ballplayer.
But there was never any doubt that a living, breathing human being resided within that athletically perfect body.
That much was on display all this week, from the moment he let you know, in barely couched terms, that it was not his decision to end his playing career on a Friday night in the Bronx with his team in playoff contention and nearly 50 more games left to play, let alone four, and possibly more, home runs left for him to hit.
It was on display when he greeted his preteen daughters, Natasha and Ella, on the field after the final out of the Yankees' 6-3 victory; he contributed an RBI double in the first inning, and the emotion of the moment had him wiping away tears.
And it was especially on display in Joe Girardi's postgame interview, when the manager broke down weeping and needed several moments to compose himself before finishing an answer on how the last week of his professional association with Rodriguez had strained their 12-year relationship.
"I think some people think that I wanted to make negative decisions, but that's not the case," Girardi said. "I have a huge heart, and if this is the last time he plays, I wanted it to be something never forgotten."
As great as Derek Jeter was, and as long as he and Girardi were associated, there was nowhere near the intensity of emotion following his final game in 2014 as there was after this one.
Part of that was due to the fact that unlike A-Rod, Jeter left on his own terms after a yearlong farewell tour in which he did virtually everything he wanted: play shortstop and bat second in the lineup practically every day.
Rodriguez had no such say in his own Yankee endgame. He was forced to sit through the humiliation of a news conference in which the team tried to sweeten the hammer blow of an unconditional release with a four-game, mini-farewell tour and the promise of a vaguely outlined job as a special adviser to the organization.
But another part of it had to do with the differences between Jeter and A-Rod, who were once portrayed as close as brothers but in reality are quite different.
Jeter was flawless in an antiseptic, almost robotic way; A-Rod was always painfully human, as evident by his first foray into PED use back in 2001, when he signed that 10-year, $252 million contract with the Texas Rangers that would come to weigh like an albatross around his neck.
He survived that one by becoming a postseason hero in 2009, after several embarrassing failures early in his Yankees career. But when he lapsed again, following hip surgery a couple of years later, it was clear that for all his incredible talent, Rodriguez was still fighting the same demons that had plagued him when he was a 25-year-old with the hopes and expectations of an entire sport on his shoulders.
That is why there was always a poignancy to Alex Rodriguez, even when you knew he was trying to con you. That is because it was obvious he was also trying to con himself.
"I know that I'm someone who loves the game tremendously, and I've made some tremendous mistakes," he said. "I've also worked extremely hard in trying to come back and do things the right way and mend certain types of relationships and it was extremely uncomfortable, making some of the phone calls I had to make. But if you look at the 22 years, to have this type of ending, the last two years is what I'm most proud of, and what I'm most going to remember."
Aside from the steroid admissions, there was a public divorce, some embarrassing tabloid fodder involving a centaur painting and photo of A-Rod kissing himself in a mirror, the mess with Cousin Yuri and the split with Jeter over some silly comments made in a magazine a long time ago. There was a falling out with Joe Torre over a humiliating lineup demotion in the 2006 playoffs.
Finally, there was a deep rift within the Yankees that split wide open during his contentious appeal of the 211-game ban baseball hit him with in 2013 for his involvement with Biogenesis. He and his attorneys decided to go scorched earth on the organization that had paid him more than $300 million.
Through it all, you always had the feeling that Rodriguez was being pulled in several directions but that what he really wanted most of all was to be part of a family.
The Yankees were that family, which is why he agreed to accept his firing "gracefully," because make no mistake, this was a firing, pure and simple.
For weeks now, Girardi has said, both tacitly and directly, he no longer felt that the best Yankees team he could put out on the field included Rodriguez.
That had to sting deeply for a man who won three MVPs and had hit more home runs than all but three players in baseball history, even if that success was tainted.
Still, in private moments with friends and media members he trusted, Rodriguez bared the hurt that the past few weeks had caused him, wondering privately why he was being singled out while others in the Yankees' lineup with numbers even worse than his were still playing every day.
It was a show of humanity that Jeter never would have revealed, and that Girardi recognized even while he was making the tough call to excise Rodriguez's name from his lineup card.
"Some people don't understand this comment, but I have a responsibility to do what I think is right to win games," said Girardi, who was put in an untenable position by the Yankees' front office, being asked to take the bullet for a decision that was hardly his alone, although he maintained publicly that it was. "It was really hard telling him on the nights he wasn't DHing. That was hard for me. Because there was a strong relationship there. We've been through a lot together. He's done a lot for this organization, and that was hard. But I hope that this was as good as it could get for him."
As ugly divorces go, the one between A-Rod and the Yankees probably ended about as well as it possibly could. Rodriguez started the last two games, and he delivered a big hit in his final one. He even trotted out to third base one last time, for the first out of the ninth inning, a show of humanity by Girardi that no doubt went a long way toward healing the wounds of the past few weeks.
But there will be no Alex Rodriguez in the Yankees' clubhouse come Saturday morning, and while that might not add up to a loss on the field with the next generation of Yankees -- Gary Sanchez, Tyler Austin, who will be called up to replace A-Rod, and soon Aaron Judge -- it will certainly add up to a big hole in their clubhouse.
"We have two jobs as baseball players," Mark Teixeira said. "One is to win games and two is to entertain. Alex won a lot of games when he was here. And damn, he was entertaining."
Entertaining, imperfect and human.
In a clubhouse and a sport and a world that is becoming increasingly impersonal, that is a quality of Alex Rodriguez that will be missed around Yankee Stadium every bit as much as anything he could do on the field.