Skating 4-on-3 in the overtime session, Henrik Zetterberg sent a hard pass from the low left circle to the slot, where Kronwall one-timed it through a screen and into the left corner of the net for the win.
Martin Hanzal lit the lamp and Ilya Bryzgalov stopped 31 shots in the loss for the Coyotes, who played their first game after splitting a pair of contests with Boston in Prague, Czech Republic last weekend to start the season.
Detroit eliminated Phoenix in seven games in the opening round of last year's playoffs. It was the first time the Coyotes were in the postseason since 2002.
The Coyotes scored with 10 seconds left in the opening period as Hanzal's wrister beat Howard for his first goal of the season.
Bryzgalov had to stop just five shots in the first period to keep Detroit off the board.
At the 7:03 mark of the second period, Detroit tied the game on Datsyuk's second goal of the season.
Howard turned aside all eight shots he saw in the middle stanza. Each goaltender turned aside six shots in the third to force the extra session.
Detroit hosts Calgary on Thursday...Phoenix plays in Anaheim on Sunday...Each team won twice in the series last season...Phoenix went 29-10-2 at home last season...Detroit went 1-for-8 on the power play while Phoenix was 0-for-4.