The Oilers blew a 3-1 lead in the third period, allowing the Red Wings to score three straight goals, but then forced overtime when Jordan Eberle's second goal went in with 39 seconds remaining in regulation.
In the shootout, Nugent-Hopkins got a shot past Detroit's Joey MacDonald and Edmonton's Devan Dubnyk made the advantage hold up by stopping Danny Cleary on the Red Wings' final try in the tie-breaker.
Sam Gagner contributed two goals and an assist, while Dubnyk finished with 35 saves for Edmonton, which improved to 4-0-1 in its last five games.
Todd Bertuzzi found the net twice for Detroit.
Conklin had allowed three goals on nine Oilers shots.