Adam Fox, who also scored, found Kakko with a backdoor pass near the left post and he redirected the puck into the net for the winner and his first multigoal game in the NHL.
Alexandar Georgiev made 30 saves for the Rangers (8-6-2), who extended their point streak to four games (3-0-1) and are 5-1-1 in their past seven games.
[WATCH: All Penguins vs. Rangers highlights]
Kakko has five goals in his past seven games. He is the first 18-year-old player in Rangers history to score two goals in a game.
"Just talking to him, the way he walks around now, there's just a whole new level of swagger to him that I hadn't seen since he got here, not only on the ice but off the ice," Rangers coach David Quinn said. "There's a comfort level he's attaining, and you can see it in his face."
Video: PIT@NYR: Kakko scores beautiful goal on breakaway
Kakko scored 18 seconds after Chris Kreider, Artemi Panarin and Tony DeAngelo finally got off the ice after opening overtime with a shift that lasted 2:06 because they were hemmed in the defensive zone.
"First couple minutes [of overtime] was so hard for us, but then we get the pass and a great pass by [Fox]," Kakko said. "That was easy for me."
Justin Schultz and Jared McCann scored, and Matt Murray made 24 saves for the Penguins (10-6-2), who played without center and leading scorer Sidney Crosby (17 points) for the first time this season because of a lower-body injury.
Crosby's status for Pittsburgh's next game, Friday against the New Jersey Devils at Prudential Center, is not yet known.
The Penguins had to rally after falling behind 2-0 in the first period. They have trailed by at least 2-0 in four straight games, going 2-1-1 in that span.
"It's the same problem, we need to start better," Penguins center Evgeni Malkin said. "We have to understand if we want to win, we have to play better. The last 40 minutes we played good, but it's a 60-minute game."
Video: PIT@NYR: McCann wires home power-play goal
Kakko scored on a breakaway at 6:42 of the first period to give the Rangers a 1-0 lead.
Fox scored from the right circle off a cross-circle pass from Panarin to make it 2-0 with 11 seconds remaining in the first.
"We weren't ready to play," Penguins coach Mike Sullivan said.
The Penguins rallied in the second. Schultz scored from the left circle at 1:13 to make it 2-1. McCann scored on the power play from the right circle at 12:33 to tie the game at 2-2.
It was the Penguins' first power-play goal since Oct. 13, ending an 0-for-28 slump.
"It's a good feeling," Sullivan said. "I was happy for the players because I know they want to score. It's been a little bit of a struggle. It wasn't the prettiest goal, but usually that's how it happens. We're going to try to build on that goal."
Video: PIT@NYR: Fox roofs wrist shot from the circle
The Penguins had the puck for the majority of the first two-plus minutes of overtime but managed one shot on goal and three shot attempts.
Kreider, Panarin and DeAngelo were able to change when Brian Dumoulin's drop pass missed Malkin and went all the way down the ice.
Malkin got the puck back into the zone, putting it wide of the net. Kakko intercepted the puck in the slot and gave it to Fox to start the rush up the ice that led to the winning goal.
"I wanted to wheel out three oxygen tanks for all three of them, but I didn't think they would allow me to do that," Quinn said. "Just a heck of a play by Fox to Kakko."
Video: EA Overtime Winner
They said it
"We're frustrated because we know we have better. That's what it comes down to. We can't keep having starts like this. We know that." -- Penguins forward Jared McCann
"When guys weather a storm [in overtime] and are able to battle for a little over two minutes you get your fresh legs and you think a little momentum there. It was a good compete out of them, they were hemmed there. Once we got the fresh legs we got going." -- Rangers defenseman Adam Fox
Need to know
Panarin's assist on Fox's goal extended his point streak to nine games (four goals, nine assists), tying his NHL career long streak. … Rangers center Mika Zibanejad (upper body) missed his seventh straight game. … In addition to Crosby, the Penguins were also missing defenseman Kris Letang and forward Patric Hornqvist, both out with lower body injuries. Letang has missed three straight games and Hornqvist four in a row.
Penguins: At the New Jersey Devils on Friday (7 p.m. ET; ESPN+, SN, MSG+, ATTSN-PT, NHL.TV)
Rangers: At the Tampa Bay Lightning on Thursday (7 p.m. ET; ESPN+, SNE, SNO, SNP, TVAS, SUN, MSG, NHL.TV)
Video: Kakko scores two including OT winner vs. Penguins