Cleary's tip-in late in the third period delivered the Game Seven victory for the Stanley Cup Champions, who will now meet the Chicago Blackhawks.
Detroit took a 2-0 lead with goals from Jiri Hudler and Darren Helm before Anaheim's Teemu Selanne cut the deficit.
Mikael Samuelsson then responded with a snap shot at 16:23 in the second period to make it 3-1.
Corey Perry and Bobby Ryan scored for Anaheim to even the contest at 3-3 in the third period before Cleary sent his team through.
Anaheim goaltender Jonas Hiller made 36 saves to cap an exceptional performance in his first postseason.
The eighth-seeded Ducks had previously upset top-seeded San Jose in the first round, and were seeking to beat the Red Wings in a playoff series for the third successive time.
