Mirco Mueller had the other goal for Everett (20-26-2), while Manraj Hayer chipped in with three assists.
Payton Lee made 18 saves in taking the loss for Vancouver (12-36-0).
The Silvertips were 2-for-5 on the power play, while the Giants were scoreless on their six chances.
The rebuilding Giants — with five 16 year olds and six 17 year olds in their lineup — dropped their fourth straight and third in as many night. The club sits last in the B.C. Division.
The Silvertips, who were also playing their third game in three days, all victories, remain in fourth place in the U.S. Division — 15 points behind the idle Tri-City Americans.
It was the seventh time the Giants have been shut out this season, but the first on home ice.
Cotton, a native of nearby West Vancouver, now has two shutouts for the season, both against the Giants. He also blanked them on Nov. 30 in Everett, Wash.
Winquist's hat trick was the first of his WHL career and gives him 21 goals on the season.
The Silvertips scored the only goal of the first period, added one more in the second and closed out the scoring with the only two goals of the final 20 minutes.
Two Giants saw scoring streaks come to an end. Cain Franson was blanked after five consecutive games, while Brendan Rouse, who was picked up at the trade deadline, was held off the scoresheet for the first time in his five games since joining the club.