Toews was able to beat Ilya Bryzgalov through the legs in the shootout, and Sharp followed with a shot that beat the goaltender stick side. Nikolai Khabibulin saved both Phoenix attempts, including a great glove save off Peter Mueller to secure the win.
Kris Versteeg and Sharp scored the goals in regulation for the Blackhawks, who snapped a three-game losing streak by starting their six-game road trip with a win. Nikolai Khabibulin stopped 36-of-38 shots for the win.
Zbynek Michalek and Shane Doan both scored for the Coyotes, who rallied from a 2-0 deficit but lost their third straight game. Bryzgalov allowed two goals on 40 shots in the loss.
Down by a goal headed into the third, Phoenix looked to its captain to tie the game. With Aaron Johnson off for holding, Doan took a cross-ice pass from Mueller and beat Khabibulin with a slap shot through traffic at 5:57.
Both teams had good chances in the overtime period, but Chicago had the best opportunity with a power play in the last 90 seconds of the extra session. The Blackhawks peppered Bryzgalov with shots, but none got through and the game went to a shootout.
The Blackhawks got on the board late in the first period. Patrick Kane went in on net, but his shot was saved by Bryzgalov. The rebound was right in front, and Versteeg popped it over the netminder's shoulder for the score at 18:02
But Chicago got its second goal before the buzzer sounded in the first. Sharp stole it from Phoenix forward Daniel Carcillo in front of the net and threw it toward Bryzgalov. The netminder wasn't ready for the shot, and the puck trickled past him for the tally with 50 seconds left on the clock.
The Coyotes converted on the power play with 13:52 left in the second. With good puck movement, Keith Yandle dished it to Michalek in the left circle, and his slap shot beat Khabibulin on the stick side.
Versteeg had a chance early in the third to put Chicago up by two, but his shot as he dove hit the left post, and it remained a one-goal game.