Ranked 87th in the world, the 34-year-old Bhat 24-22 19-21 21-11 in a one-hour contest to become the first Indian to come out trumps in this tournament.
The former national champion dug from his huge reservoir of experience to outlast the Dane, who he was facing for the first time in his long career.
The first game itself turned out to be a battle of attrition as the two shuttlers matched each other point for point.
There was nothing spectacularly distinct in any of the players' game and both seemed wary of approaching the net. The second game followed a similar pattern but it was Vittinghus, who came out trumps this time by converting one of the two game points he earned.
But the Dane lost steam in the decisive third game as the Indian reeled off 11 consecutive points to seal the issue and record a memorable triumph.