The Coyotes had their franchise-record, 10-game home winning streak snapped.
After 11 consecutive failures in the shootout, Clowe beat Ilya Bryzgalov with a backhand. That boosted San Jose to 59 points, which leads the NHL by two over Chicago and New Jersey.
It was the Sharks' seventh straight win. Dan Boyle and Kent Huskins scored in regulation for San Jose, which leads the Western Conference with 138 goals.
Backup goaltender Thomas Greiss made his first start for San Jose since Dec. 12 and stopped 34 shots.
The last three Phoenix games have all been decided in shootouts. Two of those have been losses to San Jose.