The goal came with 7 seconds showing in Detroit's eighth advantage of the game. The Coyotes had killed off the first seven power plays.
Martin Hanzal scored with 10 seconds to play in the first period to put Phoenix ahead 1-0. The Red Wings tied it when Pavel Datsyuk's wrist shot deflected off the stick of Phoenix's Lee Stempniak and dribbled between the legs of goalie Ilya Bryzgalov just over seven minutes into the second period.
Detroit enjoyed power plays for four minutes of overtime. The Red Wings had a four-on-three advantage when they scored the winning goal.
It was a rematch of the last season's first-round playoff series, won in seven games by Detroit with a decisive 6-1 victory in Glendale.
At 2:51 of overtime, Coyotes defenseman Adrian Aucoin was called for interference after slamming into the backside of Detroit's Tomas Holmstrom in front of the Phoenix net. A brief skirmish followed, with Holmstrom and Phoenix's Keith Yandle drawing penalties for roughing.
Detroit, with its nearly constant advantage, outshot the Coyotes 8-0 in overtime. Still, it appeared Phoenix, which killed a four-minute penalty in the third period, would hold the Red Wings off and force a shootout.
But Kronwall's slap shot, with assists from Henrik Zetterberg and Nicklas Lidstrom, got past Bryzgalov. It was Kronwall's second goal of the season.
The game was played in front of an announced sellout crowd of 17,125,
Detroit was without star Johan Franzen because of a concussion. Franzen, who took an elbow to the head from Mark Fistric in the Red Wings' 4-1 loss in Dallas on Thursday night, had three goals and an assist in the Red Wings' first four games.
The Coyotes scored in the waning seconds of the first period on their 12th and final shot of the frame. After two penalty kills and a failed power play by Phoenix, Wojtec Wolski sent a pass from left of the net to Hanzal, whose shot from straightaway sailed to the right of goalie Jimmy Howard with just 10 seconds left.
The game came one day after a significant development in the protracted efforts by the NHL to sell the Phoenix franchise and keep it in Arizona. The city of Glendale announced on Friday it had reached an agreement in principle on a new lease with a group headed by Chicago investor Matthew Hulsizer.
The NHL, in a statement on Saturday, said the agreement, once completed, would pave the way for "finally" selling the franchise, which the league bought in U.S. Bankruptcy Court more than a year ago. Hulsizer attended the game but didn't talk to reporters.
NOTES: Ray Whitney, signed in the offseason to boost the Coyotes' anemic power play, missed the game because of an upper body injury. ... Last year, Phoenix sold out its home opener, but only after drastically reducing ticket prices. Although there were many empty seats, this game was listed as a sellout of 17,125 with regular-priced tickets. ... As usual, the game was played in front of a huge contingent of Detroit fans.