Zachary Sapsford gave United the lead, before Jayden Meghoma drew Danny Röhl’s men level with his first senior goal. However, it was the home side who restored their lead midway through the second-half through Amar Fatah.
In the final minute of injury time, Rangers were awarded a penalty, and with regular taker James Tavernier substituted, Bajrami stepped-up to the spot and converted to see the game finish level.
From the weekend’s draw with Falkirk there were three changes to the line-up as Jayden Meghoma, Danilo and Djeidi Gassama all came in from the start in place of Max Aarons, Thelo Aasgaard and Oliver Antman, who is set to miss the next eight weeks with an injury.
Rangers started with Jack Butland in goal behind a back four of James Tavernier, Nasser Djiga, Emmanuel Fernandez and Jayden Meghoma.
Nico Raskin and Connor Barron retained their places in midfield, with Danilo, Djeidi Gassama, Nedim Bajram and Bojan Miovski completing the line-up.
Rangers got the action underway at Tannadice and after weathering a few early corners from the home side, it was Gers who threatened first. Tavernier’s cross was headed down in the box by Miovski before falling for Danilo but his shot was cleared by Panutche Camara.
However, with 11 minutes played, Zachary Sapsford found the opener for Dundee United. Picking the ball up in his own half, the striker turned away from Emmanuel Fernandez before driving towards the Rangers goal and finishing beyond Butland from a tight angle.
Möller came close to doubling their advantage just one minute later as he glanced a header just wide of the post as the home side took confidence from their early breakthrough.
It was an open start and David Richards pulled off a fantastic save in the 17th minute to deny Bojan Miovski. A good ball was delivered into the box by Nedim Bajrami and the Gers striker got a touch to send it goalwards, but Richards matched it with a strong hand to push away the effort.
Nico Raskin was next to see a header cleared off the line by Iurie Iovu as he headed a James Tavernier corner towards goal.
With 23 minutes played Rangers were back in it and it was a fantastic strike from Jayden Meghoma with his first goal for the club and his first in senior football. From a Rangers free-kick, the ball broke for the 19-year-old at least 25-yards from goal and he released an incredible low and powerful strike into the back of the net.
Röhl’s men were continuing to threaten as Barron was next to have a go as he sent a strike just high of the target, before at the other end Amar Fatah drove towards the goal before carving out space to get a shot away but his low strike was gathered comfortably by Jack Butland.
Two minutes into the second-half, Rangers almost had the lead for the first time this evening. Nedim Bajrami’s right-footed shot was punched away by Richards, with the rebound falling for Danlio who sent an overhead kick back towards goal and the Brazilian was incredibly unlucky as it came back off the crossbar.
Gassama had a shot saved from inside the box as Rangers continued to push for a second goal at Tannadice.
Another huge chance followed as the game passed the hour mark. Nico Raskin played a lovely ball into the box into the path of Gassama who’s shot was blocked from close range with Danilo unable to get the connection he was looking for on the rebound.
At the other end Dolček delivered a cross into the box with Watters getting a shot away which was blocked by bodies in the box before Butland saved a header from Keresztes.
From the resulting corner, Dundee United were back ahead with Amar Fatah weaving his way around the edge of the box before sending a right-footed strike low into the back of the net.
With just over 10 minutes remaining, the Rangers head coach turned to his bench with a triple change. Captain James Tavernier, Nico Raskin and Danilo were replaced by Max Aarons, Mohamed Diomande and Youssef Chermiti.
Meghoma was denied a second of the evening with a good save from David Richards as he blocked his low shot with his feet.
Another change followed as Thelo Aasgaard was introduced for the final few minutes in place of Bojan Miovski.
With six minutes added time, Rangers were still looking to rescue something from this fixture and they were awarded a penalty in the 96th minute after a foul on Max Aarons inside the box.
Penalty taker James Tavernier had been substituted earlier in the match, Nedim Bajrami stepped up to the spot and made no mistake in converting the penalty with his first Scottish Premiership goal of the season.
RANGERS: Butland, Tavernier (Aarons, 77), Djiga, Fernandez, Meghoma, Raskin (Diomande, 77), Barron, Danilo (Chermiti, 77), Gassama, Bajrami, Miovski (Aasgaard, 90).
Subs: Kelly, Nsiala, Sterling, Cameron, Curtis.
Dundee United: Richards, Esselink, Iovu, Ševelj, Camara (Strain, 54), Sapsford (Dolček, 62), Ferry, Fatah (Trapanovski, 77), Stephenson, Keresztes, Möller (Watters, 62).
Subs: Adams, Strain, Graham, Trapanovski, Sibbald, Stirton, Watters, Cleall-Harding.