Miller entered the game at the beginning of the third period and made 20 saves after Gibson sustained a lower-body injury late in the second.
Jakob Silfverberg and Josh Manson scored for the Ducks, who won their third straight game and are 5-1-1 in their past seven.
WATCH: [All Ducks vs. Golden Knights highlights]
"There's been some good hockey in our end and we want to keep it going," Miller said. "There have been some close games all the way through that we didn't find ways to win. This one was the same, and we found a way to come through."
Anaheim (30-20-11, 71 points) leads the Calgary Flames and Los Angeles Kings by two points for third place in the Pacific Division.
Marc-Andre Fleury made 18 saves for the Golden Knights, who lost for the first time in three games and were shut out at home for the second time this season (Nov. 28, Dallas Stars).
Vegas (39-16-4) has 82 points, one more than the Tampa Bay Lightning for the NHL lead, three ahead of the Winnipeg Jets and Nashville Predators for first place in the Western Conference, and 10 more than the San Jose Sharks, who are second in the Pacific Division.
"[The Ducks] worked hard, had a gritty game," Golden Knights coach Gerard Gallant said. "The game plan they wanted to play was low shot totals, slow it down, goalies freezing pucks. I didn't think we played with the same passion in the first half of the game."