The Yankees trailed 4-0 after six innings. Rodriguez hit his 695th career homer in the seventh and Beltran added his own two-run shot in the eighth.
For Beltran, it was his 18th of the season, one shy of his total from 2015, and the latest in a line of big hits for the 39-year-old outfielder.
"Homers? I don't really look for that. That's not my approach," Beltran said. "My approach is to stay in the middle of the field. It's great I'm able to do that. But my whole approach is being able to hopefully stay in the field and give my team a chance to get the rally going."
The Yankees scored three times in the ninth for a 7-4 lead. Closer Aroldis Chapman gave up back-to-back homers to Eduardo Escobar and Kurt Suzuki with two outs in the bottom half before holding on for his 13th save in 14 chances.
It was 4-all when Fernando Abad (1-1) walked Chase Headley to lead off the Yankees. Didi Gregorius slipped a bunt past Abad and beat second baseman Brian Dozier's throw to put two runners on with nobody out.
After a wild pitch, Brett Gardner hit a sacrifice fly.
"You want to feel that you can always come back. You want to have a team that grinds out at-bats," Yankees manager Joe Girardi said. "We have guys that have been around a long time and have seen relievers, and it's not the first time they've seen certain guys."
Byung Ho Park homered and had a sacrifice fly, giving Minnesota a 4-0 lead.
Andrew Miller (4-0) pitched a scoreless eighth for the win.
Rodriguez hit his eighth home run this year, connecting off Ricky Nolasco one pitch after Beltran beat out an infield dribbler for a single.
Beltran tied it with an opposite-field homer off Kevin Jepsen. Gardner narrowly beat out a grounder to short to set up Beltran's shot.
Twins manager Paul Molitor had a chance to force the switch-hitting Beltran to bat right-handed if he'd brought in Abad, his lefty setup man.
Molitor declined to second-guess his decision to stick with Jepsen, who lost his closer job earlier this month.
"Beltran is having a great year. He's dangerous both ways, but I'd rather see him hit left-handed, in that situation, than right-handed against a little more experience," Molitor said. "Kevin got ahead of him, but then he left a changeup up that (Beltran) took out the other way."
Yankees starter Michael Pineda tied his season high with nine strikeouts.
Park made it 3-0 in the fourth inning with his 12th homer, a two-run shot to right. He added a sacrifice fly in the sixth.
RARE DAY FOR CHAPMAN
Chapman hadn't allowed any home runs in his first 17 games this season. The last time any team tagged him for consecutive homers was on May 19, 2013, when the Phillies' Erik Kratz and Freddy Galvis did it.
"When that happens, you can't lose your concentration," Chapman said through a translator. "Your job is to get the next guy out, and that's it."
Yankees: 1B Mark Teixeira (torn cartilage, right knee) is close to heading to the minor leagues for a rehab assignment, Girardi said. Teixeira, who has been sidelined since June 4, ran the bases before Saturday's game and could be ready to start his rehab stint as early as Tuesday. "Every test, he has passed so far, so we're looking at a couple more days' work and then sending him out," Girardi said.
Twins: GM Terry Ryan said OF Miguel Sano (strained left hamstring) did a significant amount of running on Friday and will undergo agility drills on Monday. Ryan indicated that Sano could be ready to head out for a minor league rehab assignment next week.
Yankees: RHP Nathan Eovaldi (6-3) went 6-0 over a span of seven starts, but he hasn't won since May 29. He's posted a 9.82 ERA and given up five home runs in his last three outings. He is coming off his shortest start of the year, in which he gave up six runs in four innings in a loss at Colorado on Tuesday.
Twins: RHP Ervin Santana (1-7) is 0-5 with a 7.71 ERA in his last five starts. He won his first four career decisions against the Yankees, but he hasn't beaten them in 10 starts since 2008.
Shaky Royals bullpen hangs on for 7-5 win over Blue Jays
— Jakob Junis ran his consecutive scoreless streak to 14 innings as the Royals staked him to a seven-run lead Thursday night, and a shaky Kansas City bullpen held on after giving most of it back for a 7-5 win over the Toronto Blue Jays.
What goes up must come down ... except this foul ball
Willy Adames skies a foul ball behind the plate, but the ball gets stuck on the catwalk at Tropicana Field.
Guerrero's home run inches Blue Jays closer to Royals
Vladimir Guerrero Jr.'s home run lands a whopping 458 feet out in left-center field as the Blue Jays pull within two runs of the Royals.
Royals jump ahead in first few innings and never look back
The Royals take advantage of an offensive attack in the first four innings and never look back in a win vs. the Blue Jays.
Garcia's first career HR comes at the perfect time for Texas
Adolis Garcia launches a 2-run shot in the 10th inning in the Rangers' win over the Rays.
Rays tie game after Rangers commit two errors on one play
Jose Trevino's pickoff attempt to second goes into the outfield, then Leody Taveras' throw to third hits the runner and gets away to allow the Rays to tie the game.