The Tangerines have won 10 of those matches and a ruthless counter-attack that ended with Obadeyi rounding goalkeeper Greg Fleming was enough at Somerset Park.
United rode their luck though, with Ayr United creating but failing to convert several first-half chances.
Jamie Adams missed the best of those when he blasted over from two yards.
The Tangerines' are now level on 34 points at the top of the Championship and Ray McKinnon's men have kept five consecutive clean sheets.
United's goal-making move started with an Ayr free-kick which Gary Harkins could not finish.
The visitors quickly sent a direct ball forward to Obadeyi, who with a combination of pace, skill and a lucky break of the ball found himself with the opportunity to run the ball past Fleming and stroke into an empty net.
Ayr United responded well, Harkins and Paul Cairney shooting over at the end of some nice football.
Adams' bad miss came at the end of another impressive move - Ross Docherty finding the marauding Nicky Devlin, whose low cross looked to be on a plate for Adams, but he scooped it over.
At the end of a poor second half the hosts were ruing those chances wasted in the first.
Ayr United manager Ian McCall: "Their goal came from us nearly scoring and we got isolated one to one. Nicky Devlin, who was outstanding, went to ground.
"But after that we had chance after chance - I think we had about 18 corners to their two and something like 60% possession. We were the better team throughout."
Dundee United manager Ray McKinnon: "We scored early and I don't know if that upset us or put us off our game, but we certainly weren't at our best today.
"The character to win the game was very important. We probably dropped below what I expect from the guys in terms of on the ball, but certainly off the ball we worked extremely hard and that's a real positive."
Tope Obadeyi's solitary strike against Ayr United helped Dundee United extend their unbeaten run to 12 games.