Both sides had been forced into changes by injury or suspenion, but they were quickly into their stride as play flowed from end to end.
However, it took 21 minutes for either goalkeeper to be tested, when it needed a smart double-handed save from United goalkeeper Dave Richards to deny Karlsson's shot from range.
Just when the Dons started to apply some sustained pressure, United pounced to take the lead on the break.
Fatah dispossessed Nicky Devlin and advanced into the penalty box before side-footing low past goalkeeper Dimitar Mitov into the far corner for the on-loan Troyes forward's third goal of the season.
A fresh-air clearance from Bert Esselink allowed Graeme Shinnie to race clear inside the penalty box shortly after the break.
The restored Aberdeen captain, though, could not mark his 400th Premiership appearance with a goal as his shot was charged down by Richards.
United were rarely seen as an attacking force, but Zac Sapsford should have done better on the break when the striker raced clear but miscued his shot.
Aberdeen made United pay for that miss as Richards could only parry Adil Aouchiche's low drive and on-loan Bologna midfielder Karlsson pounced to fire the equaliser high past the goalkeeper for his sixth goal of the season.
That re-awakened United as an attacking force, but neither side could find a winner.