The visitor spurned a number of chances in the first half and gave Brondby its first goal in the Europa League this season as Balogun headed into his own net just before the break.
Rangers struggled to find their rhythm in the early stages of the second period, but Steven Gerrard's positive changes paid dividends as Hagi raced in behind to lash home an equaliser in the 77th minute.
The result leaves the defending Scottish champion level on points with second-placed Sparta Prague, who were beaten 3-0 by Lyon, while Brondby remains bottom of Group A.
Rangers started strongly and looked threatening in the first 10 minutes, with Alfredo Morelos seeing a header ruled out and then firing over after good work in the box before Fashion Sakala fizzed a low shot just wide.
Morelos came close again with a long-range effort before Joe Aribo headed onto the roof of the net, but it was the hosts who took the lead on the stroke of half-time as Josip Radosevic flicked a corner on and Balogun inadvertently nodded past Allan McGregor.
The visitor pushed for an equaliser and threw men forward as Gerrard made several second-half changes. Balogun attempted to make amends for his earlier mistake with a well-struck shot on the bounce and Glen Kamara blasted at the near post from distance, but both were denied.
However, Hagi, who was introduced 10 minutes into the second half, latched onto a pass from fellow substitute Ryan Kent and thumped a low drive into the bottom-left corner to keep Rangers' qualification hopes alive.