A looping header from Ryan Astley shortly into the second half stunned Ibrox as the Gers' task was made all the more difficult by the dismissal of Nasser Djiga on the hour-mark.
However, the Gers ensured a share of the spoils in stoppage time when the captain stroked home from the penalty spot after Gassama was brought down inside the Dundee penalty area.
Drama ensued moments later as Cyriel Dessers slammed home what he thought was the winning goal, but a subsequent VAR check curbed the Rangers celebrations as attention now shifts to Tuesday evening's UEFA Champions League second leg qualifier in Plzen.
There wasn’t much to shout about in the opening exchanges as Rangers moved from side-to-side in pursuit of an avenue through a Dundee defensive block defined by its five-man backline.
Indeed, it took until a quarter of an hour for the Gers to ask questions of their Tayside visitors as Lyall Cameron, a Dundee academy graduate, looked to combine with Cyriel Dessers but the duo weren’t quite on the same wavelength.
Heads were scratched all over Ibrox when Djeidi Gassama was released down the left-hand side by Jefte but, after picking up a similar shooting position to where he broke the deadlock in midweek, he elected to square across the penalty area to Dessers who was eventually crowded out.
Joe Rothwell then tested his luck from the edge of the penalty area after being teed up by Cameron before Gassama blasted off target himself moments later after he was picked out by Oliver Antman.
Dundee’s attacking forays had been few and far between in the first half, but they went closest to opening the scoring when Clark Robertson nodded back across goal and off the upright with Jack Butland stranded.
The crack of the woodwork appeared to spark the visitors into action as the Dundee captain was once again bemoaning the frame of the goal when he headed onto the post again just minutes later, although the offside flag was later raised against the defender.
Rangers had found the net in each of their last 16 league home tussles against the Tayside outfit coming into this evening, but any signs of a first half breakthrough appeared slim with Dundee goalkeeper Jon McCracken having yet to truly muddy his jersey.
Flashing beyond the Dundee goal was a thunderous driven effort from Jefte four minutes shy of the interval as the Gers looked to spark some momentum into their attacking endeavours.
The reverse pass inside the full-back had been the theme for Rangers in midweek and it almost paid dividends twice in quick succession shortly after the restart.
First, Jefte was initially smothered by the onrushing McCracken after being released beyond the Dundee backline before Gassama showed nimble footwork to carve his way into the visiting penalty area from the left-hand side and pick out Mohamed Diomande whose first-time effort was thwarted.
However, it was Steven Pressley’s side who took the lead just moments later when a searching free-kick deep to the far post was nodded back across by Robertson and headed smartly over a helpless Butland from Ryan Astley.
The hosts sought an immediate response as the lively Jefte stood up an enticing cross for Cameron to attack in the penalty area, but the midfielder couldn’t quite divert his header on target.
And yet the task got even more difficult for Russell Martin’s side on the stroke of the hour mark when Nasser Djiga was shown a straight red card for clipping the heels of Joe Westley as the Dundee man raced through on goal.
With a man disadvantage the onus was on the hosts to find a route back into the contest without compromising themselves further defensively.
Souttar went close from a corner before the Gers probed with purpose and James Tavernier, on as a substitute, curled wide from distance.
Further attacking impetus was added by Martin as Findlay Curtis replaced Jefte as, at the opposite end of the pitch, Nico Raskin did tremendously well to nip possession away from Finlay Robertson as Dundee midfielder broke forward into space.
Antman then unleashed at the visiting goal after weaving his way inside but, in keeping with the theme of the evening, it was comfortable for McCracken.
Rangers found an equaliser as the game entered stoppage time when Gassama was fouled after showing terrific footwork on the byline and Tavernier, as he so often does, dispatched the subsequent penalty with aplomb.
Dessers slammed home what he thought was the winner minutes later, but VAR came to the visitors' rescue.
RANGERS: Butland, Aarons (Danilo, 63), Souttar, Djiga, Jefte (Curtis, 76), Rothwell (Tavernier, 63), Diomande (Dowell, 55), Cameron (Raskin, 55), Antman, Gassama, Dessers
Subs not used: Kelly, Cortes, Barron, Fernandez
DUNDEE: McCracken, C. Robertson, Astley, Wright, Digby, F. Robertson, Hay (Acquah, 81), Hines-Samuels (Donnelly, 90), Westley (Yogane, 66), Graham, Jones (Koumetio, 90)
Subs not used: Carson, Ingram, Galabov, Allan, Crombie
REFEREE: Don Robertson