Ajeti blazed over from two yards and Celtic's day might have been worse, with United adamant they should have had a first-half penalty when Cameron Carter-Vickers bundled over Nicky Clark.
Celtic took the lead early on through Liel Abada, but Ian Harkes levelled two minutes later to keep United one place above their hosts, who stay sixth in the table, six points behind leaders Rangers.
David Turnbull, Tom Rogic and Jota all hit the bar for Celtic but they failed to reproduce the form that had seen them score 26 goals in seven consecutive home wins, and have now dropped more points than they have gained in the league.
Ange Postecoglou was without eight players, most of them attackers, and the only player he brought into the side, Josip Juranovic, limped off in the 26th minute.
Celtic had early chances. Ajeti headed over and Juranovic had a strike palmed away by Benjamin Siegrist.
United had their own fitness issues with Charlie Mulgrew and Marc McNulty dropping out, but it was quickly evident they had plans to get forward in numbers whenever possible.
Celtic took the lead in the 16th minute when Abada got in front of his marker to head home Jota's cross from six yards.
But United levelled with a similar goal. Ilmari Niskanen emerged with the ball on the left wing following Anthony Ralston's sliding challenge and whipped in a cross which Harkes headed into the corner of the net from six yards.
Celtic should have hit back instantly from the same source as their opener. Abada brought down Jota's far-post cross and squared for Ajeti but the striker blazed over from two yards right in the middle of the posts.
The next big talking point soon came after Celtic lost possession on the edge of their box.
Carl Starfelt did not look comfortable receiving a Rogic pass and was robbed by Peter Pawlett before the ball fell for Clark. Carter-Vickers appeared to barge the forward on the back and Clark lost his footing, and the United players and bench were stunned when referee Kevin Clancy played on.
Adam Montgomery replaced Juranovic and initially struggled with his passing, although he was far from the only Celtic player to do so.
Jota continued to be a threat, though, and saw efforts turned past by Siegrist and Ryan Edwards.
Turnbull almost lifted an audibly frustrated home support in the final minute of the half when he turned and unleashed a powerful left-footed strike from 25 yards that hit the underside of the bar. Ajeti tucked home the rebound but he had pushed Edwards to get to the ball and the goal was rightly disallowed.
Ismaila Soro came on for the second half for James McCarthy, who had picked up a knock following a late tackle from Jeando Fuchs, but also generally struggled to inject the tempo required by Postecoglou.
Abada shot over twice as Celtic broke but United threatened in between when Niskanen curled just wide.
Soro took time to get up to speed, but Celtic began to push United back for longer spells.
United defended their box well but there were openings. The unmarked Ajeti missed what looked another good chance when he failed to get clean contact on Abada's cross. Jota volleyed just over under pressure and Rogic headed against the bar from eight yards.
Louis Appere forced a good stop from Joe Hart as United threatened a late winner before Jota hit a looping strike off the bar ahead of the full-time whistle, which was met by loud jeers.