Hampus Lindholm and Adam Henrique scored, and Gibson made 29 saves for the Ducks (1-2-2), who went 0-for-5 on the power play to fall to 0-for-12 on the season.
"Our execution got better as the game went on," Ducks coach Dallas Eakins said. "We just need a little bit more finish now."
Rantanen scored off a centering pass from rookie defenseman Bowen Byram to give Colorado a 2-1 lead at 6:20 of the third period.
It was the first point in the NHL for Byram, the fourth pick in the 2019 NHL Draft who made his NHL debut on Thursday in a 4-2 loss at the Los Angeles Kings.
"They spent almost the whole entire time with their shifts in the offensive zone and creating scoring chances, and then they finally get rewarded on an unbelievable pass from Byram to Mikko," Avalanche coach Jared Bednar said. "They played a huge factor and really elevated their game in the third period."
Henrique came back and scored off a centering pass from Danton Heinen to tie it 2-2 at 7:42.
"That was a big goal," Eakins said.
Gibson saved a penalty shot by Nathan MacKinnon at 8:29 of the third to keep the score tied.
"It was like a whole bunch of air exhaled at the same time on the bench," Eakins said. "That's a dangerous, dangerous player and [Gibson] handled it like it was next to nothing. That's a massive save at an absolute critical part of the game."
The Ducks committed a minor penalty in the opening minute of the game for the second straight game, and the Avalanche capitalized with the goal by Joonas Donskoi for a 1-0 lead at 2:56.
Avalanche defenseman Samuel Girard had room to bring the puck into the left circle and pass to Donskoi on his right. He scored on the one-timer with three seconds left on the power play.
Colorado's power play is 10-for-25 this season.
Grubauer made 18 saves in the first period, including a right pad save on Jakob Silfverberg from in close and then a partial stick save on his follow-up attempt. Byram then cleared Silfverberg's second shot as it slid in the crease toward the goal line.
"Tonight, I didn't think we deserved to win, but Grubauer was outstanding," Bednar said.
Lindholm scored 47 seconds into the second period on a slap shot from the inside edge of the left circle to tie it 1-1.
Anaheim took possession on the forecheck and Rickard Rakell passed the puck back to Lindholm, who scored from the inside edge of the left circle.
The Ducks and Avalanche play again on Sunday in Anaheim.
NOTES: Landeskog (465 points; 201 goals, 264 assists) passed Dale Hunter to move into ninth place on the Avalanche/Quebec Nordiques scoring list. … Colorado defenseman Erik Johnson needs more conditioning before he'll be ready to play, Bednar said. Johnson was scratched shortly before the season opener on Jan. 13 because of a positive COVID-19 test. ... Avalanche forward Martin Kaut was activated from the taxi squad and made his season debut. He played 5:50. ... Ducks forward Sonny Milano missed the game with an upper-body injury and is day to day. Ducks forward Max Jones was recalled from the San Diego of the American Hockey League and made his season debut. He had two hits in 12:28 of ice time.
