The 2010s were a difficult decade for the Gers, with financial problems resulting in their demotion to the third tier in 2012, but they gradually built their way back up and on Wednesday clinched their place in the round of 16 of the competition for only the second time since finishing as runners-up in 2008, when it was known as the UEFA Cup.
Rangers' immense comeback at Ibrox in the first leg put them in control of the tie, having battled back from 2-0 down to win 3-2 with a trio of goals in the final 23 minutes.
Although the Scottish club ceded possession for much of the encounter, they attacked with greater purpose and were well worth the lead when Ryan Kent finally netted in the 61st minute.
Despite Braga's undoubted threat, Rangers looked far more menacing going forward and arguably should have scored twice in the first 20 minutes – Matheus thwarted Florian Kamberi after a Scott Arfield-led break, while Kent shot wide.
However, their best chance in the first half came when Raul Silva clumsily handled inside the box, only for Matheus to produce an excellent save to keep out Ianis Hagi's spot-kick.
Failing to heed the previous warning signs about their extremely high defensive line, Braga eventually fell behind just past the hour – Hagi playing a long pass over the hosts' backline and Kent finished well after holding off Fransergio.
Braga piled on the pressure towards the end, but aside from a Paulinho header hitting the foot of the post, they struggled to trouble Rangers, who had a 77th-minute Arfield goal disallowed for offside.