Evan Rodrigues beat Ilya Sorokin with a backhander for the only goal in the shootout, spoiling a 46-save effort for the Islanders' goaltender. Avalanche forward Mikko Rantanen had a career-high 10 shots on goal.
''I feel like I have good deception and good hands and I feel like if I sell it hard enough I can get a goalie to buy it and it's worked out for me,'' Rodrigues said.
The shutout was Georgiev's second of the season and 10th of his career.
New York completed the Western Conference portion of its five-game road trip, which wraps up at Madison Square Garden against the Rangers on Thursday.
The playoff-like matchup was in contrast to the teams' first meeting, a 5-4 home win for the Islanders on Oct. 29.
''You kind of got the feeling the way both goalies were playing that the first goal would probably be the difference-maker,'' Avalanche coach Jared Bednar said.
Martin Kaut, recalled from Colorado of the AHL earlier Monday, nearly scored on the Avalanche's first power-play chance, early in the second period. Though Colorado didn't score on the man advantage, the momentum shifted after that.
The Avalanche had a second opportunity midway through the second when Zach Parise went off for tripping. Colorado had three quality shots on Sorokin but he came up with big saves.
The Avalanche outshot New York 15-7 in the second and continued to carry the play in the third. Val Nichushkin and Rodrigues had prime chances midway through the period that Sorokin turned away.
Sorokin bounced back from allowing five goals on 29 shots in his start at Arizona on Friday night.
''There were a couple of pretty odd bounces in Arizona that you can't do anything about,'' Parise said. ''He's played great and he played great again tonight.''
Colorado forward Andrew Cogliano went to the locker room late in the third period after hitting the boards in the New York zone. Cogliano's right skate got tangled with Parise's left skate and he went into the boards with his right shoulder.
''We'll see how he presents in the morning and we'll have a better idea on the timeline,'' Bednar said.
Avalanche defenseman Cale Makar earned integrity points with an unselfish gesture. Late in the first period Makar went down behind his net as Mathew Barzal tried to get the puck from him with his stick. The referee put up his arm indicating a delayed penalty on Barzal but Makar waved his glove to tell the ref he slipped and a penalty wasn't warranted.
''I think I would have felt a lot less guilty if I didn't say anything than if I did,'' Makar said. ''I apologize to (my teammates). I don't plan on doing that again.''
Makar may get good-natured grief from his teammates but he got respect from the Islanders.
''Just an honest player who wants to beat you fair and square,'' New York coach Lane Lambert said.
The Avalanche acquired F Denis Malgin from Toronto in exchange for F Dryden Hunt. ... New York G Semyon Varlamov is day-to-day with a lower-body injury suffered in the third period of Saturday's 5-2 win at Vegas. ... Colorado's Kurtis MacDermid returned to the lineup after missing 16 games with a lower-body injury.
Islanders: At the New York Rangers on Thursday night.
Avalanche: Host Montreal on Wednesday night.
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