Gibson stalled the league's fourth-highest scoring team, beating Minnesota and former coach Bruce Boudreau for the first time in three meetings.
The Wild were 0 for 5 on the power play and lost in regulation for the first time in six games and the fifth time since the start of December, a span of 34 games. Devan Dubnyk made 22 saves.
Cramarossa scored for the first time in 16 games 4:38 into the first period. Corey Perry's shot from Dubnyk's left side was stopped, but the puck dropped near Dubnyk's feet and Cramarossa alertly tapped it in.
Following the goal, Anaheim went nearly 10 minutes without a shot. Gibson had the cushion he needed, though.
Minnesota had 13 shots on goal in the first period and then peppered Gibson in the second with another 16 shots. The Wild outshot the Ducks 37-23 in the game.
Closing against Anaheim has proven to be a tough task for any team this season. The Ducks have allowed the fewest third-period goals in the NHL this season.
Minnesota had plenty of chances. The closest was Christian Folin's slap shot early in the third that deflected hard off the post behind Gibson. The Wild were shutout for the second time at home this season.
Minnesota had scored four power-play goals in the past two games and had 13 in its previous 13 home games. The Wild were a league-best 30.1 percent on the power play at home this season heading into the game and sixth overall.
Ducks forward Antoine Vermette was ejected in the third period with a game misconduct for abuse of an official.