Celtic trail leaders Rangers by six points having played two games fewer.
United had impressively subdued Neil Lennon's side, who were playing their first league fixture in 13 days after having two games postponed for defender Boli Bolingoli's quarantine transgression.
But the late blow leaves Micky Mellon's Premiership newcomers fifth, level on points with Celtic.
It was a tale of Celtic frustration for much of this Tannadice encounter as United - with four players aged 22 or under - energetically harassed the visitors out of their stride.
Whenever Celtic finally breached the wall of tangerine, they found Siegrist in inspired form.
After Edouard thumped the outside of the upright from Mark Reynolds' attempted clearance, Siegrist saved smartly from Ryan Christie and Edouard.
The theme continued after the restart, with Christie thwarted as United's Swiss goalkeeper clawed away a shot that looked destined for the top corner.
Jamie Robson had United's first effort on target just shy of 70 minutes, but the respite was brief.
Celtic kept pushing and substitute James Forrest fired inches over, Callum McGregor drilled wide and Edouard grazed the bar with a free-kick.
Their perseverance finally paid off when Siegrist again foiled Christie, but was helpless as Ajeti collected the rebound to fire past three despairing United players on the goal line.
Man of the match - Benjamin Siegrist
Defeat was cruel on the Dundee United goalkeeper, whose heroics looked to have secured a point for his sideWhat did we learn?
Celtic looked better suited to a front two. Ajeti's introduction off the bench gave sole striker Edouard a partner up front and led to the crucial breakthrough. Ajeti didn't come cheap at 4.5m from West Ham United, but the 23-year-old's contribution in this tight Tannadice tussle could prove priceless months down the line.
United, meanwhile, already look an accomplished Premiership side just five games in. Mellon has them well drilled and disciplined, with a blend of youth and experience that should keep them clear of any relegation worries.
What did they say?
Dundee Utd manager Micky Mellon: "I'm pleased with my players and very frustrated at not getting something from the game. It gives us something to work from. We have a lot of talented young players with great desire."
Celtic manager Neil Lennon: "The players' attitude was tremendous, they didn't get frustrated and kept going. We got the goal we thoroughly deserved - we were outstanding at times. I think 1-0 flattered Dundee United."