The home side went into the game low on confidence after they suffered a virtual humiliation at the hands of Vejle in a 0-4 defeat in the previous round.
However, OB had a bright start to the game when Rasmus Falk produced a superb through-ball to Noah Ganaus to open the score after just five minutes.
Randers didn't take long to respond when the Odense offside trap failed to work leaving Mohamed Touré with an easy job to square the ball for the onrushing Norman Campbell who was left with an easy tap-in to restore parity.
Randers then took the lead through a similar scenario when Campbell turned provider for Florian Danho to put the visitors in control of the match.
Against the run of play, the home side then got themselves back into the game when they were awarded a penalty when a shot from Nicolas Bürgy hit Wessel Dammers on the arm and Jann-Fiete Arp made no mistake from the 11-metre spot.
The end-to-end action continues and OB then managed to restore the lead when Fiete Arp produced a wonderful delivery for Jay Roy Grot to make it 3-2 for the home side.
Randers pushed for the equaliser throughout the second half but couldn't find a way through against a resilient Odense rearguard.