The hosts struck the first blow in the contest when James Tavernier converted from the penalty spot on the stroke of the half-time interval and they appeared to be heading in the direction of victory when Rodrigo Zalazar was given his marching orders for the visitors.
However, the Portuguese side struck back when Gabri Martinez converted with a little over quarter of an hour remaining to ensure a share of the spoils on a blustery evening in Govan.
The Gers boss named three changes to his side from the victory over Livingston at the weekend, with Youssef Chermiti returning from injury and coming into the starting lineup alongside Max Aarons and Connor Barron.
There was also a first appearance in the matchday squad for Academy midfielder Calum Adamson, who has impressed on loan at Alloa Athletic this season as part of the club’s co-operation agreement with the League One outfit.
Braga are no strangers to Govan having visited in two separate memorable evenings in the knockout stages of this competition in 2020 and 2022.
Common to both those contests has been an attacking endeavour from the Portuguese side, and they were first to offer a glimpse at goal when Ricardo Horta played a give-and-go on the edge of the penalty area with the talented Zalazar and drove a powerful effort across goal that Jack Butland was equal to down low.
Zalazar is a player much talked about in European circles, and he was at the heart of another dangerous Braga moment minutes later when he swivelled smartly and stabbed an outside-of-the-boot pass through to Horta, but it was fortunate for the hosts that the forward’s first-touch deserted him and Butland sprung from the goalline to smother.
The visitors were offering the greater attacking threat in the opening half-hour, albeit they were fleeting in working the Gers goal, and they almost engineered an attempt at goal from a short corner routine when Gustaf Lagerbielke stretched to nod Horta’s delivery goalwards but it was out of the reach of the defender.
Still, the most dangerous opportunity belonged to the hosts when Mohamed Diomande spun intelligently from a direct James Tavernier pass. The Ivorian slid in Danilo in acres of space to his left, but the Brazilian contrived to curl a tame effort straight down the throat of Lukas Hornicek with only the Braga goalkeeper to beat.
Rangers were beginning to show signs of life as Hornicek then beat Djeidi Gassama’s lasered cross as far as Tavernier who lashed over from distance.
There were cries for a spot-kick moments later when Gassama fell to the turf when challenging for a header at the far post, the decision of no penalty checked and cleared by VAR, before Danilo headed down in search of Chermiti, only for the Portuguese forward to be crowded out by Hornicek and a defender.
Drama then ensued on the stroke of half-time as the hosts broke the deadlock. Tavernier’s corner delivery appeared innocuous, but it was only on the advice of VAR that the referee was advised to take a trip to the pitchside monitor where the replays showed the ball striking Fran Navarro’s outstretched arm.
Who else but to step up in a crucial moment but the captain as he sent Hornicek the wrong way to bring Ibrox to its feet.
Rangers were certainly keen to continue their momentum into the second period as, just seven minutes after the restart, Gassama was released down the right-hand side and his cross fell to Danilo who was unable to connect.
The hosts then worked a corner delivery back into the penalty area when Danilo floated a delicious delivery across the face of goal and Chermiti, lurking at the far post, steered goalwards before Lagerbielke hacked the ball off the goalline.
There was a further sting in the tale for the Portuguese visitors on the hour-mark as they were reduced to ten men for a petulant moment from Zalazar when he motioned his head into Nico Raskin after the pair clashed midway inside the Braga half.
However, the visitors struck when the chips were down with 22 minutes remaining to level the contest as Nasser Djiga failed to deal with Victor Gomez’s hopeful delivery before Gabri Martinez profited to stroke a composed finish beyond Butland at the far post.
Djiga almost made amends a few minutes later when Manny Fernandez done well to meet a corner at the far post and nod back across goal, but the on-loan Wolves defender glanced wide of the Braga goal.
Moments later, Tavernier headed straight at Hornicek from a Gassama cross before Diomande combined with substitute Thelo Aasgaard but slipped in the process of pulling the trigger from the edge of the penalty area.
Braga came close to stealing an improbable victory in stoppage time when Butland flapped at an inswinging corner and had to be alert to deny Gomez's effort which bounced up off the surface.
Diomande was then shown a second yellow card in stoppage time to even up the numbers before the referee's full-time whistle drew a frenetic contest to a conclusion.
RANGERS: Butland, Tavernier, Fernandez, Djiga, Aarons (Meghoma, 88), Barron (Miovski, 76), Raskin, Diomande, Gassama, Danilo (Aasgaard, 69), Chermiti (Antman, 88)
Subs not used: Kelly, Wright, Cameron, Adamson, Curtis
BRAGA: Hornicek, Gomez, Lagerbielke, Niakate, Arrey-Mbi, Moutinho (Grillitsch, 65), Zalazar, Gorby (Moscardo, 90), Martinez (Carvalho, 81), Horta, Navarro (El Ouazzani, 45)
Subs not used: Sa, Bellaarouch, Lelo, Oliviera, Victor, Dorgeles, Rodrigues, Vidigal