The only goal of the game was scored when German playmaker Jann-Fiete Arp converted a penalty after just under an hour of play.
With the win, OB have now not lost in their last six matches and the Odense outfit will now aim to hang on to their advantage as they travel to Aarhus on Sunday.
Thursday's match in Odense was even with chances at both ends. AGF came close to taking the lead several times in the first half.
AGF full-back Rasmus Carstensen in particular, who has otherwise shown a goal-scoring ability of late, missed a great opportunity after just a few minutes.
Fifteen minutes later, Carstensen had another chance completely free in front of goal, but he kicked the ball straight at OB goalkeeper Theo Sander.
OB also had several chances in the first 45 minutes, but all of them missed the target.
There is no VAR in the cup competition. And that was perhaps a good thing for the home side who won a very dubious penalty when Jann-Fiete Arp fell very lightly to the ground in the box under the challenge of Erik Karl, but the referee did not hesitate to point to the spot.
Fiete Arp converted the penalty kick chipping the ball into the middle of the goal, while AGF goalkeeper Jesper Hansen threw himself to one side.
AGF tried to force an equalizer, but couldn't find a way past goalkeeper Theo Sander, who kept the goal clean for the third successive time in the cup competition.
