The Croatians took the lead after 10 minutes as Duje Cop found himself unmarked in the penalty area. The striker making no mistake with a simple finish.
Maribor found a crucial away goal through a fortunate source on 39 minutes, as Dinamo captain Milan Badelj turned a low free kick into his own net.
Badelj atoned for his error late on to give Dinamo a deserved victory. The skipper broke forward to slide home Cop's through ball 15 minutes from time.
Maribor are still very much in the tie despite a disappointing performance as attention turns to next week's second leg in Slovenia.