Having missed the goalless first leg at Anfield three weeks ago through suspension, Virgil van Dijk's long pass led to Mane's excellent first-half goal at the Allianz Arena on Wednesday.
"He's on fire at the minute and let's hope it continues for the rest of the season."
Dutch centre-back Van Dijk then headed last season's runners-up Liverpool into a 2-1 lead from a Milner corner on 69 minutes after Joel Matip's own goal had handed Bayern a first-half equaliser.
Mane made absolutely sure of Liverpool's place in the quarter-final draw by nodding in six minutes from time.
Liverpool's 3-1 aggregate victory means they join Tottenham Hotspur, Manchester City and Manchester United in the quarter-finals, completing a clean sweep of English sides reaching the last eight for the first time since 2008/09.
"It is good for English football -- that's what you want," added Milner.
"We have good feelings from last year, so hopefully we can carry that on and go as far as we can."
It is a bleaker picture in Germany as Bayern's exit leaves no Bundesliga club in the quarters for the first time since 2005/06, as Borussia Dortmund and Schalke also bowed out in the last 16 -- to Tottenham and Man City respectively.
This is the first time since 2011 that Bayern have failed to make the quarter-finals.
"Both teams were very cautious for a long period, but the second goal took the belief out of us," said Bayern defender Mats Hummels.
"Jurgen Klopp is good at neutralising the opponents' strengths and he did that once again.
"We didn't play well enough to create enough clear chances at this level."
- Henderson forced off -
Milner was preferred to Fabinho in Liverpool's midfield, but Jurgen Klopp was forced into an early change when captain Jordan Henderson hobbled off on 12 minutes. Brazilian Fabinho made an early entrance and Milner took the skipper's armband.
Klopp had urged his side to "be brave on the ball" and it was that, combined with an amazing piece of skill by Mane, which gave them the lead.
Mane spun away and looped his shot inside the far post to give the visitors a valuable away goal in the 26th minute.
The bravery Klopp wanted flowed through the Liverpool ranks and Milner, winning the midfield battle, set Firmino free in the box, but the Brazilian blasted his shot at Neuer.
The ball went in off Matip, who was focused on marking the Polish striker, to the despair of the Liverpool centre-back and the delight of the home crowd.
The tempo of the game dropped sharply after the restart, but Bayern wasted a chance on the hour mark when Gnabry fired in a cross, only for Lewandowski to mistime his slide.
Liverpool were heading for the quarter-finals when Milner swung in the second of two corners in quick succession and Van Dijk powered a header past Neuer.
Mane scored his second when Mohamed Salah's chip found him at the far post to head home his sixth goal in four matches, as Liverpool's travelling fans started singing
"You'll Never Walk Alone", while the Bayern supporters began to slip away.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)