Los Blancos held a 1-0 lead from the first leg of their last-16 clash last month, and a 1-1 draw at Bernabeu was enough for Carlo Ancelotti's side to advance to the final eight.
Vinicius Junior sent Real Madrid ahead in the 65th minute of the contest before Willi Orban levelled the scores three minutes later, but Leipzig could not find a second which would have forced extra-time, as the 14-time European Cup winners progressed to the quarter-finals.
The first half-chance of the match came for Vinicius in the 12th minute, but the Brazil international could only divert a cross from Dani Carvajal over the crossbar.
Leipzig had an opening of their own three minutes later, with Lois Openda breaking into a dangerous position, but the striker's effort from distance was always moving wide of the post, before Orban missed the target with a header from a Xavi Simons free kick.
The visitors had another opportunity in the 16th minute through Openda following an excellent pass from Dani Olmo, but the 24-year-old, who was carrying the fight for the visitors early on, again missed the target.
Real Madrid were finding it difficult to carve out openings, with Leipzig defending impressively against a front three of Vinicius, Jude Bellingham and Federico Valverde, with Aurelien Tchouameni, Toni Kroos and Eduardo Camavinga then operating in the midfield positions.
Tchouameni did well to block a strike from Simons in the 39th minute of the contest, with Leipzig continuing to cause problems at Bernabeu, and Real Madrid goalkeeper Andriy Lunin was then forced into action to keep out a curling effort from Simons in the closing stages of the first period.
Openda came close again before the break, striking just wide following a Leipzig corner, with the visitors proving to be the more productive outfit in the final third of the field in the opening 45 minutes, but Real Madrid's one-goal lead in the tie still stood at the interval.
Having seen his side fail to register a shot on target in the first half, Real Madrid head coach Ancelotti made a change at the break, with Rodrygo replacing Camavinga.
There was more purpose about Real Madrid at the start of the second half, and the breakthrough on the night arrived in the 65th minute, with Bellingham breaking towards the penalty area before releasing Vinicius, who picked out the back of the net with an excellent finish.
Leipzig responded in the 68th minute, though, with Orban heading a cross from David Raum into the back of the net to make it 2-1 on aggregate.
Lunin made a smart save to keep out a strike from Benjamin Sesko in the 74th minute, with Leipzig looking to level the tie, before Openda headed wide of the target from a free kick two minutes later.
Olmo and Benjamin Henrichs both had efforts in the period that followed, but the former saw his strike miss the target, before the latter was denied by Lunin in the 80th minute.
Both managers made changes in the spell that followed, and Leipzig continued to commit players forward in a desperate attempt to score a second goal which would have forced extra-time.
Leipzig goalkeeper Peter Gulacsi made a smart save to deny Rodrygo in the 89th minute of the contest, with the match then heading for four additional minutes.
Olmo came so close to scoring a second for Leipzig in stoppage time, but the Spain international struck the woodwork, with Real Madrid surviving the dangerous moment to edge their way into the final eight at the expense of a talented Leipzig outfit.