Phoenix got goals from five different players and 31 saves from Ilya Bryzgalov to level the series 3-3.
The Coyotes will host Game Seven on Tuesday with a chance to advance to the second round for the first time since the franchise moved to Phoenix.
Facing elimination, Lauri Korpikoski opened the scoring for the visitors at 4:10 in the first period and Mathieu Schneider gave the Coyotes a 2-0 lead in the second.
Darren Helm added a final goal for Detroit with 3:31 remaining.
The two-time defending Western Conference champion Red Wings had won the previous two games by a combined score of 7-1 to take control of the series but could lose in the opening round for the first time since the 2005-06 season.
(Reporting by Jahmal Corner in Los Angeles, editing by Ed Osmond)