Both teams emerged with a 2-2 record in the young season, Phoenix winning on the road for the first time in three attempts.
Detroit's two goals came 53 seconds apart early in the second period from Joakim Andersson and Jonathan Ericsson, giving the Red Wings a 2-1 lead.
Mike Smith then shut down Detroit the rest of the way and Mikkel Boedker tied the contest just 64 seconds after Ericsson had put the Red Wings in front.
Smith made 28 saves while Howard gave up three goals on 37 shots.
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