As a result, Giovanni van Bronckhorst's side will face RB Leipzig in the final four, after the German side saw off Atalanta.
The match got off to a ferocious start when Rangers took the lead just 70 seconds into the clash as Borsa Barisic's cross founds its way to James Tavernier who slid in at the far post to level the tie.
It appeared that the Scottish champions had doubled their lead just minutes later when Kemar Roofe slammed home, however the referee adjudged that a handball had occurred in the build-up, leading to the goal being chalked off.
Rangers eventually took an aggregate lead just before the break when Tavernier added his second of the night from the spot, but Braga wouldn't give up.
Just when it looked like Rangers' dominance would be rewarded with a semi-final berth, David Carmo levelled the score on aggregate with seven minutes to play.
However, Roofe finally got his goal, and it was worth the wait, as it sent the Scottish champions past Braga and into the final four.
Rangers deserved huge credit for their thunderous start, but that was largely down to the electric atmosphere at Ibrox ahead of kick-off.
Fans of the Gers have been looking forward to a European occasion of such magnitude for some time and the chance to compete for a spot in the final four of the Europa League is the sort of occasion that the Glasgow side relish.
As Tavernier's effort hit the back of the net, former Rangers star Charlie Adam was quick to give his reaction during talkSPORT 2's live coverage, saying: "Welcome to Ibrox!
"What a game, what a start for Rangers."
While Roofe's effort was ruled out moments later, it certainly wasn't enough to stop the hosts from dominating the first half.
Tavernier and Roofe both went close with efforts inside the first 25 minutes of the game, while the former Leeds striker went close again after 32 minutes when his close-range effort rattled the crossbar in front of the home fans.
As half-time approached, Rangers needed to turn the dominance into a lead, and they did exactly that shortly before the break when Roofe was bundled over in the area by Vitor Tormena who was sent off for a professional foul.
Tavernier stepped up and sent the goalkeeper the wrong way to put the hosts 2-1 up on aggregate going into the break.
The second half started in much the same way, with Joe Aribo's decent effort sent just wide of the mark from inside the penalty area, before Roofe had another goal ruled out midway through the half after being flagged for offside.
Despite Rangers' dominance, 10-man Braga managed to find their way back into the clash when David Carmo headed home from a corner to level the tie and take it to extra-time.
While anxiety levels were building, there was no need as Roofe finally got the goal that he deserved after converting from close range following a fine ball from Tavernier down the right.
The tie was all-but wrapped up at the midway point in extra-time when Iuri Medeiros was also sent off for Braga, meaning that the Portuguese side would contest the final 15 minutes with a two-person disadvantage.
However with the game done and dusted, those at Ibrox can now look forward to a two-legged tie with RB Leipzig where they will be able to test themselves against one of the best sides in the competition.