Rantanen took a cross-ice pass from Nathan MacKinnon and scored with a shot from the right face-off dot.
Colorado went on the power play 44 seconds into overtime after Vegas forward Reilly Smith was called for slashing Rantanen's stick out of his hands.
"It's all over the League that if you slash the other guy's stick out of his hands or break his stick, it's always a penalty," Rantanen said. "I'm a pretty strong guy too, so I can hold on to my stick, but that was a pretty hard slash. I don't think I would purposely drop my stick in the [defensive zone]."
Golden Knights coach Peter DeBoer didn't like the penalty.
"Ah, just a soft call, but I can't even blame the refs," DeBoer said. "They're fighting through the embellishment of grabbing your face or falling down or dropping your stick every period. So, you know, I can't even blame the referee on it. They fooled them on it."
Brandon Saad and Tyson Jost scored, Samuel Girard had two assists, and Philipp Grubauer made 39 saves for the Avalanche, the No. 1 seed in the Honda West Division, who won 7-1 in Game 1 on Sunday.
"'Grubi' bailed us out today," Rantanen said. "He had a really, really good game. He kept us in the game. He is basically the reason why we got to OT, and our special teams stepped up there, but Grubi was awesome for sure, First Star of the game."
Alec Martinez and Smith scored, Shea Theodore had two assists, and Marc-Andre Fleury made 22 saves for the Golden Knights, the No. 2 seed.
"It's not easy to take when you lose in the playoffs," DeBoer said. "I thought we deserved better. I really liked a lot of the things we did and our effort but got to find a way to bury one of those chances and win a game. They held serve at home; we've got to do the same."