Phoenix defenseman Sami Lepisto scored 29 seconds into the game, the fastest playoff goal in franchise history, and Wojtek Wolski also scored late in the second period.
Ilya Bryzgalov made 29 saves for the Coyotes .
Detroit's Valtteri Filppula made it 1-all late in the first period and Johan Franzen also scored for the Red Wings .
Jimmy Howard made 29 saves for Detroit.
Detroit will host Game 4 on Tuesday night, needing a win to avoid being on the brink of elimination after advancing to the past two Stanley Cup finals.
The Coyotes quickly quieted a fired-up crowd at Joe Louis Arena when Lepisto got the third shot of the game past Howard, scoring the quickest goal for the franchise since it was in Winnipeg.
Filppula's power-play goal got the fans back into it at the 14:42 mark of the first period, but the Red Wings failed to score in the second period and they allowed Wolski to score with 32 seconds left in the second.
Phoenix looked faster than Detroit at times, beating veterans such as defenseman Nicklas Lidstrom to the front of the net and getting to pucks in the corners quicker.
The Red Wings had what appeared to be two goals waved off, including one by Todd Bertuzzi with 3:02 left in the game.
Coyotes center Shane Doan had what appeared to be an upper-body injury in the second period and did not return. Doan collided with Howard on his first shift of the second period, lost his balance and ran into the endboards. He later walked toward the dressing room and didn't play the rest of the period.
NOTES: Doan had a goal and an assist during the first two games, which were split in Phoenix. He led the Coyotes with 55 points during the regular season. ... Two octopi were thrown on the ice during the national anthem, a playoff tradition in Detroit, then building manager Al Sobotka let one of them slip out of his hand as he twirled it in what was perhaps a bad omen. ... Phoenix F Vernon Fiddler was scratched after needing five stitches in his upper lip after getting hit by a stick in Game 2. ... Wolski has scored in each game against Detroit. ... Detroit F Tomas Holmstrom played in his 155th game, moving past Gordie Howe for seventh place on the postseason games list for the Red Wings .