Detroit twice came from a goal behind in the second period only to have the Coyotes go back in front less than a minute later.
The Red Wings then tied the game for a third time in the second stanza on a slap shot by Valtteri Filppula and they went in front for the first time on Justin Abdelkader's wrist shot 2:32 into the third.
Shane Doan tied it at 4-4 with a power-play goal midway through the third, but the Red Wings produced the final three scores of the contest -- two of them from Zetterberg and another from Filppula.
With the best-of-seven series tied at 1-1, Detroit will host Game 3 on Sunday.
Both goaltenders struggled, but Detroit's Jimmy Howard eventually prevailed by stopping 27-of-31 shots. It was the first post-season victory for Howard.
Ilya Bryzgalov was beaten six times on 38 shots.
It was the highest scoring game to date in this year's playoffs. Ottawa and Philadelphia combined for nine goals in the first game of their series.