Gers centre-back Leon Balogun headed into his own net following a corner in the final seconds of the first half after Steven Gerrard's side had squandered several chances.
It was the home side's first goal of the group stages, but Rangers increased the tempo in the second period and Hagi's fine finish in the 77th minute from a pass from fellow substitute Ryan Kent earned the draw.
The point keeps Rangers in third place in Group A, level on four points with Sparta Prague, who are second on their head-to-head record and who visit Ibrox on 25 November.
The Scottish champions have to face the Czech side at home and travel to leaders Lyon in their final two fixtures so still have work to do to qualify for the knockout stages, but how they needed this result in Denmark.
The Scottish Premiership leaders went into the game boosted by a 6-1 thrashing of Motherwell at Fir Park on Sunday.
Fashion Sakala, who scored a hat-trick against Well, kept his place, but Gerrard brought in left-back Borna Barisic for Calvin Bassey as attacker Kent returned to the bench for the first time since picking up a hamstring injury in September.
The Danish champions had goalkeeper Mads Hermansen and top scorer Mikael Uhre both back from injury, but Rangers might have scored three times in the first nine minutes.
Alfredo Morelos headed into the net from a James Tavernier corner in the fifth minute, but the ball had gone out before it reached the Colombia striker.
Morelos fired a shot over the bar minutes later after controlling a Barisic cross inside the box before Sakala drove a shot wide after racing through the middle.
The home side worked themselves into the game and the Rangers goal survived a couple of anxious moments.
However, it was still the visitors who looked the more likely to break the deadlock, with Morelos, Connor Goldson and Joe Aribo all having efforts of various quality, none of which, however, unduly troubled Hermansen.
Then, with seconds remaining in the first half, a Josip Radosevic corner from the left was helped on by Christian Cappis and the ball came off the head of the hapless Balogun and went past keeper Allan McGregor to change the complexion of the match.
Rangers' start to the second half was uninspiring and in the 53rd minute Kevin Tshiembe's close-range shot was blocked by Scott Arfield for a corner which was defended uneasily.
Gerrard had seen enough and on came Kent, Kemar Roofe and Hagi for Morelos, Sakala and Arfield.
The Light Blues stepped up the pressure and a powerful Balogun strike in the 61st minute went straight to Hermansen.
Steven Davis made way for Juninho Bacuna in the 71st minute, but it was Kent and Hagi who combined, with the latter firing low past Hermansen and into the far corner to at last bring Rangers level.
With Bassey on for Barisic, the Ibrox side and Brondby went looking for the winner, but after a frenetic ending both sides had to settle for a point.
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