Thirteen minutes after the break, Rangers were level on aggregate thanks to Antonio Colak’s header before Malik Tillman completed the comeback (78) to send the hosts through to the play-off round.
There was brief concern before Tillman’s winner when James Sands was shown a second yellow card for a last-ditch challenge, but the decision was overruled by the linesman.
And it was the visitors who would end up finishing the match with 10 men when Lazare Amani was sent off with seconds to go as Rangers secured a memorable victory.
“We know we are capable of delivering big nights for the club. But still you have to do it over and over again,” Van Bronckhurst said.
How Rangers created history
Rangers had never overcome a 2-0 defeat away from home in the first leg of a European tie before this match. Some of last season’s games at Ibrox when they reached the Europa League final were epic and this one reached similar heights.
There was excitement before the game around the return of Gers winger Ryan Kent from an ankle knock, with left-back Borna Barisic and midfielder Scott Arfield also back while Ridvan Yilmaz, Scott Wright and Steven Davis dropped to the bench.
Alfredo Morelos, who came off the bench on Saturday to score the clincher in a 2-0 win against Kilmarnock after a five-month absence due to a thigh injury, started again among the substitutes but there was a feeling the Colombia striker would be required at some point.
Willed on by their enthusiastic but occasionally anxious fans, Rangers hastily tried to unsettle USG, who were unchanged from the first leg.
Tillman knocked a Tom Lawrence corner past the near post before Colak headed a free-kick from Tavernier over the bar.
The visitors, meanwhile, barely left their own half but with their first corner in the 19th minute, defender Siebe Van der Heyden glanced Simon Adingra’s delivery just past the far post and Ibrox breathed a sigh of relief.
In the 28th minute Union Saint-Gilloise keeper Anthony Moris reacted quickly to tip a header from Colak over the bar, from a terrific Lawrence free-kick, and the visitors held out again.
It looked like they would see out the first half but they were undone in the 44th minute when referee Anastasios Sidiropoulos judged that Barisic’s deep cross into the box had struck the arm of Van Der Heyden and Tavernier duly sent Moris the wrong way with his spot-kick and the Champions League dream was on.
Rangers pressed again from the start of the second half and again USG filed back into defence looking to spring on the counter.
The second goal came when Lawrence’s raking pass from the left found Tavernier inside the box and he cut the ball back to Arfield. The midfielder’s drive from 16 yards was parried by Moris and Colak was first to the loose ball to head in from a yard out as Ibrox exploded in noise and colour again
USG were rocking and moments later Moris had to dive to his right to push away a decent drive from Gers midfielder John Lundstram.
Rabbi Matondo replaced Arfield in the 64th minute and the game began to stretch with spaces opening up.
USG tried to emerge from their shell and Light Blues keeper Jon McLaughlin raced from his goal to thwart Adingra, who was put through by Loic Lapoussin.
Then came Rangers’ third as Barisic’s looping cross into the box saw Tillman rise early and above Moris to head into the empty net.
Ben Davies and Morelos immediately replaced James Sands and Colak and Moris rushed from his goal to deny Morelos 35 yards out after the striker had been sent clear by Kent, whose cross from the left minutes later just evaded Morelos.
Amani, booked in the 75th minute, was sent off after picking up a second yellow to complete an extraordinary night in Govan.
‘Passion, desire and fire’ delights Van Bronckhorst
Rangers manager Giovanni van Bronckhorst:
“We needed a big performance and the pressure was on us after last week.
“We know we are capable of delivering big nights for the club. But still you have to do it over and over again.
“That’s why I am so proud of my players, the way they played, with passion, with desire, with fire to go to the next round.
“That is the only way you have to be on the pitch, to create the atmosphere we created with the fans today.
“It was an amazing night for all of us.”
Van Bronckhorst admitted his heart was in his mouth at 2-0 in the second half when referee Anastasios Sidiropoulos showed defender Sands a second yellow card for a challenge, before changing his mind after consulting his linesman.
The Dutchman, who was unhappy with a penalty awarded to USG in last week’s 2-0 defeat in Belgium, said: “We knew last week wasn’t a penalty, which made it harder for us.
“Today, I was happy with the decisions of the referee, of course it was in our favour.
“There was one moment in the second half with James Sands, I was like, ‘this can be a very important moment for the game’. It was good that he didn’t give the yellow card and gave the ball to us.
“We want to be in the Champions League and I saw the desire in my players.”
The former Feyenoord player and boss looked forward to playing against PSV, coached by compatriot Ruud van Nistelrooy.
He said: “It is a familiar team for me. It is great to see Ruud van Nistelrooy, again there are so many familiar faces.
“I was assistant to Fred Rutten (PSV assistant) at Feyenoord a couple of years ago.
“It will be nice to be back in Holland again after so many months in Glasgow – but we will prepare well.
“It is a big chance to take the last hurdle and play Champions League football.”
Geraerts: Penalty was turning point
Union Saint-Gilloise boss Karel Geraerts:
“Yes, until that moment, everything was under control.
“We were aware of the first 20 minutes because everyone said they would start really hard.
“We controlled everything really well and when you lose a goal it is hard to get the same scenario in the second half.
“The second goal gave the opponent more confidence and the third goal…there was a problem in the defence but they made the most of the chance.”