The Coyotes twice took a one-goal lead, then fell behind by one before Radim Vrbata forced overtime on the power play with 6:25 to go in regulation.
San Jose eventually won its third in a row when Couture and Marleau both beat Mike Smith in the shootout. Mikkel Boedker scored for Phoenix in the first round, but Antti Niemi stopped the other two attempts to clinch the win.
Niemi was credited with 33 saves while Smith stopped 28 shots. Phoenix has lost two in a row.