His time on the podium was made to wait, however, as he was issued a random drug test immediately after the game.
Enter Spurs coach Gregg Popovich.
"Kawhi wanted it badly," Popovich said, "and he went in and took it."
Leonard scored 39 points, capping the performance with a 6.5-second sequence of brilliance on both ends of the floor to key the 112-110 victory and elevate his standing in the debate over the NBA's Most Valuable Player award.
Then in an interesting but unrelated twist, the NBA tabbed Leonard for a drug screening, and the forward took a little longer than expected to provide a sample because of slight dehydration. That, in turn, delayed Leonard's availability to the media.
The NBA tests players randomly four times during the season and twice during the offseason.
Leonard's game-high 39 was matched by Houston MVP candidate James Harden in a battle that went to the wire, as the former knocked down a 25-foot go-ahead 3-pointer with 25.4 seconds remaining before blocking the latter's shot on the ensuing possession with 18.9 ticks left to seal the game.
"Those guys were hitting shots, big shots," Leonard said.
So his mentality on the final sequence was to "just get back and get a stop," he said.
Even Harden complimented Leonard's block, although he said he got fouled on the play, apparently by Spurs forward David Lee.
"That was a good block, but initially I got fouled," Harden said. "That's why the ball took so long to get up, but it's a good block."
"I knew he was going to get [the block] because James took that little bump. So that's too long," Parker said. "I saw Kawhi coming, and I knew he was going to get it."
Monday's performance marked Leonard's fourth consecutive outing in which he scored 30 points or more. Leonard has now produced 23 outings this season in which he has scored 30 points or more, joining George Gervin (seven), David Robinson (four) and Tim Duncan (one) as the only players in franchise history to put together 20 such performances for the Spurs in one season.
What Leonard does on the other end of the court, though, is what sets him apart, Popovich said.
"The block [at the end of the game] is what makes him special," the coach said. "Obviously, the 3, you know Harden makes 3s. Kawhi makes 3s. This guy makes 3s, that guy makes 3s. Steph [Curry] makes 3s. Everybody does that. But I don't know who goes to the other end and does what he does -- not that many people on a consistent basis, and entire game, game after game."
The Rockets led the Spurs by as many as 16 points. In fact, San Antonio's 16-point deficit was its largest at home after the first quarter in exactly six years to the day.
Parker drilled a trio of 3-pointers to key a 14-2 run by San Antonio in the third quarter that ended on a Lee dunk that cut Houston's lead to two points (69-67) with 5:23 left in the quarter. By the 3:30 mark of the quarter, another Lee dunk gave the Spurs their first lead since the 9:31 mark of the opening frame.
After the game-deciding sequence, Leonard drew a foul and hit both free throws with 14.5 seconds left to put the game further out of reach.
Leonard played at least 40 minutes in back-to-back overtime victories on back-to-back nights in different cities before taking Sunday off to rest, only to put in 39 minutes against the Rockets.
Ginobili later explained his look of disbelief upon witnessing Leonard's last-second heroics.
"I was kind of in shock. Not because he hasn't done it," Ginobili said. "He has made already quite a few of those. But, the whole situation: the way he played in the fourth quarter; that shot was crazy, because he was contested and off the dribble, a very tough shot to make. And then getting that block with Harden coming full speed is very hard to do, very hard to do. You've got to be very gifted, and he is.
"He's very confident now, and you can see it. Before, he was more introverted and waiting for things to happen. Then, he started to realize the type of talent he has and how important he is for the team. And, he knows he's going to have the ball. So, he's even trusting more teammates, waiting a couple plays and knowing we're going to play for him.
"That's something your confidence gives you. First, in your talent, and then confidence that the team is going to look for you because they need you. When you reach that point, it changes a lot because your mindset is completely different. So, he's at that point he knows he owns the team -- that we need him to do what he's doing to win, or to have a chance -- and that puts you in a great mental spot."
As tensions heightened down the stretch, Leonard called off Danny Green, who was guarding Harden, on defensive switches because he wanted to match up with the Rockets guard. Green said he tried to guard Harden to give Leonard a breather, but the Spurs forward "wouldn't have it."
Harden connected on only 1 of 5 shots in the fourth quarter for four points, while Leonard hit 4 of 6 for 17 points, including 2-of-2 from long range and 7-of-7 from the free throw line.
Leonard scored 25 points in the second half and 15 of San Antonio's final 19 points in the final five minutes.
Still, Leonard refuses to get caught up in comparisons and the MVP conversation.
"I don't know how they pick or choose it," Leonard said. "I don't really think about how other guys play the game. I'm just playing basketball, trying to make my team the best team right now coming down this stretch. I'm just trying to win the game. I'm just buying into myself, and just trying to be the best defender on that end, and then coming down on the offensive end and trying to be the best player."
Popovich also declined to engage in the MVP talk.
"We don't talk about MVP awards. We don't talk about any awards," Popovich said. "We don't talk about championships. We just play the game. Everybody tries to do what they can to make the team better, and whatever records or awards or wins that come, that's the way it is."