But while their dreams of reaching next month’s final in Seville are still very much alive, there will be frustration nonetheless in the Ibrox camp at the manner in which they found themselves with a deficit to overturn.
Having defended intelligently and robustly for most of the night at the RB Arena, Rangers were undone by an 85th-minute set piece which they could not resist sufficiently to prevent Angelino scoring with a stunning strike.
It sets up another potentially epic European night at Ibrox next Thursday when Rangers will hope home advantage can prove decisive. But they remain the underdogs against a slick Leipzig side who are especially effective on the road in Europe, having won away from home against Atalanta and Real Sociedad in their two previous ties in the knockout stages.
Van Bronckhorst may be an instinctively attack-minded coach but circumstances dictated that he adopted a far more circumspect approach on this occasion.
Indeed, the formation he deployed would surely have earned a wry nod of approval from the late, great Walter Smith. It certainly had echoes of the ultra-pragmatism which the legendary Ibrox manager utilised throughout Rangers’ last run to a European final back in 2008.
John Lundstram joined Connor Goldson and Calvin Bassey in a three-man central defence flanked by full-backs James Tavernier and Borna Barisic who both had to curb their normal enthusiasm for getting forward.
Ahead of the defensive unit, a hard-grafting midfield trio of Ryan Jack, Joe Aribo and Glen Kamara offered additional protection.
Aribo was clearly expected to try and break forward and fill a ‘false nine’ role in support of Ryan Kent and Scott Wright whenever possible, but those opportunities were few and far between as Leipzig dominated.
Rangers were under pressure from the off and initially seemed a little nervy, perhaps understandable given the magnitude of the tie.
They survived an early and highly optimistic claim for a penalty when Christopher Nkunku tumbled under a challenge from Goldson but gradually settled to their task well, restricting the number of clear-cut openings Leipzig were able to carve out.
It was a night when the Rangers players needed to combine bravery with awareness in and around their own penalty area and Bassey set the tone for their defiance with a tremendous block to deny Nkunku after a poor attempt from Barisic to clear the danger.
Goldson produced an equally well-timed intervention to prevent the lively Konrad Laimer from getting a shot on target as Leipzig continued to probe.
Rangers were scarcely able to get out of their own half but Wright did manage to get them briefly on the front foot when he forced a corner after half an hour. Barisic picked out Goldson with the set piece but the defender’s header failed to trouble the underworked Peter Gulacsi in the Leipzig goal.
Leipzig were quickly back on the attack with Tavernier the next Rangers player required to make a crucial block inside his own penalty area to repel a powerful shot from Angelino.
While Rangers saw out the rest of the first half without any further significant threat being posed by Leipzig, van Bronckhorst decided a change in strategy was required at the start of the second half.
Rangers switched to a 5-4-1 formation, Aribo moving forward as a lone front man while Kent and Wright dropped into midfield.
It immediately sparked a more progressive spell from the visitors as they caused the Leipzig defence real anxiety for the first time.
Kent broke free down the left and his low, driven ball across the face of the six-yard box had just too much pace on it for either Aribo or Tavernier to connect.
When Wright and Aribo then combined on the right, Tavernier was presented with a shooting opportunity which was deflected over.
Leipzig responded as Nkunku forced Allan McGregor into the first save of the match, the veteran goalkeeper dealing comfortably with a powerful strike which flew straight at him.
Rangers made their first change when Wright was replaced by Fashion Sakala, providing an injection of pace and energy for the final stages. But it also seemed to take some of the momentum out of their play with the Zambian international unable to hold the ball up as effectively as Aribo had been doing since the interval.
There was a real let-off for the visitors in the 70th minute when Nkunku was guilty of a horrible miss, taking the ball around McGregor but unfathomably slashing his shot wide of the target.
When Nkunku then headed over from close range, it seemed as if Leipzig were fated to be frustrated in their efforts to secure an advantage to take with them to Glasgow next week.
But Rangers were finally breached by Angelino’s moment of top quality with five minutes of regulation time remaining. McGregor did well to touch a shot from Tyler Adams over the top but from the resulting corner kick, swung over from the right by Nkunku, Sakala’s headed attempt to clear saw the ball drop to the unmarked Angelino whose sweet first-time volley arrowed into the corner of the net.
Rangers managed to negotiate the remaining minutes without suffering any further damage, leaving the tie tantalisingly in the balance.