The Scottish giants made two changes from their 3-2 win in the first-leg, with Samuel Edmundson slotting at centre-back while Florian Kamberi replaced the injured Jermain Defoe.
Kamberi had a glorious chance to give Steven Gerrard's charges an early lead in the eighth minute after going clear on goal but his shot was saved by Braga goalkeeper Matheus.
The visitors had another squandered another chance on 19 minutes after Ianis Hagi fashioned a cut-back for Ryan Kent, who could not capitalise after firing wide of a gaping net/
Despite seeing a lot of the ball in the opening exchanges, Braga only had their first clear chance after 25 minutes, as Paulinho's headerwas tipped over the bar by Rangers shot-stopper Allan McGregor.
Gerrard's troops were rewarded with a penalty on the stroke of half-time after Raul Silva handled the ball inside the box, with the on-field decision upheld by the Video Assistant Referee. However, Hagi failed to take advantage as his spot-kick was saved saved by Mathews.
After a quiet start to the second half, Rangers broke the deadlock on the hour-mark after Kent raced clear from a through ball over the defence and struck a low shot into the bottom corner to make it 1-0 on the night.
The Portguese side upped their intesity in attack after that and came closing to finding the equaliser on the night with 20 minutes left after another Paulinho header came off the outside of the upright.
Rangers thought they had bagged a seond goal with 13 minutes left after Scott Arfield finished off Connor Goldson's header that came off the post but the goal was disallowed for offside.
Second-half substitute Sheyi Ojo almost got the second goal for Rangers in stoppage time with a low curling shot but Matheus was on hand to make the save as the encounter finished 1-0 on the night and 4-2 on aggregate, sending the Scottish side through to the last 16.