Vatrano cut it to 2-1 at 15:15 with a one-timer from the high slot before Adam Henrique tied it 2-2 at 18:09, scoring on a breakaway after Cam Fowler poked the puck away from Rantanen at the Ducks' blue line.
"We could have mailed it in here [after we] went down by two goals," Vatrano said. "But we had the will to come back."
Vatrano then gave Anaheim a 3-2 lead at 5:18 of the third, scoring on a wrist shot from the high slot after he spun away from Samuel Girard, and Brett Leason pushed it to 4-2 at 7:25 when his tip-in from atop the crease went in off Girard.
"[The Ducks] know they're out of a playoff spot. Regardless of that, they're still fighting for their jobs for next year," Avalanche forward Logan O'Connor said. "Lots of young and hungry guys, guys that are looking to make a splash at the end of the season."
Rantanen cut it to 4-3 at 12:09 with a one-timer from the right circle on a power play. It was his 54th goal of the season, tying Joe Sakic for the franchise record since the team moved to Colorado for the start of the 1995-96 season.
"They have a powerhouse team," Henrique said. "If you go out there and just float around, you'll get embarrassed."