Just four days after City thrashed Liverpool 4-0 to seemingly lay down a marker next season's title race, both sides showed why there has been a chasm between them in the Reds' romp to a first league title in 30 years.
Liverpool were far from their best in their first outing at an empty Anfield as champions, but still ground out the three points in a manner they so often have this season in the final 20 minutes.
Mane broke struggling Villa's stubborn resistence by smashing Naby Keita's pass in off the underside of the crossbar from close range for his 20th goal of the season before substitute Curtis Jones rounded off a big weekend for the 19-year-old after signing a new five-year contract on Saturday.
Victory moved Liverpool onto 89 points and they need four wins from their remaining five games to surpass City's record 100-point total from 2017-18.
"We are where we are because we win difficult games," Klopp said.
"It was not a brilliant performance but it was a good performance that gets us three points."
Villa remain in the relegation zone, one point from safety as they battle to avoid an immediate return to the Championship.
City, by contrast, saw another dominant display go unrewarded as Che Adams's spectacular first goal for Southampton claimed a 1-0 win for the Saints.
Pep Guardiola's men had 74 percent possession and had 26 shots but could not find a way past Alex McCarthy as Fernandinho hit the post and Gabriel Jesus, David Silva and Bernardo Silva all wasted huge chances.
"It was a little bit of a resume of this season - one mistake and they punished us," said Guardiola. "We played a really good performance but it was not enough."
- West Ham blow leads -
At the other end of the table, West Ham edged closer to survival despite twice blowing a lead to draw 2-2 at Newcastle.
Shock winners against Chelsea in midweek, West Ham were ahead within four minutes at St James' Park when Michail Antonio pounced after Jamaal Lascelles failed to cut out Jarrod Bowen's cross.
Newcastle were level after 17 minutes as the influential Allan Saint-Maximin slipped in Emil Krafth and his cross was converted by Miguel Almiron.
Tomas Soucek restored West Ham's lead when he lashed home after Declan Rice headed Ryan Fredericks' corner against the crossbar.
But Newcastle equalised 92 seconds later as Jonjo Shelvey beat the West Ham offside trap, took Dwight Gayle's pass and beat Fabianski from close range.
After earning their first away point since December, West Ham are four points clear of the relegation zone with five games left to guarantee their top-tier status.
John Egan's first top-flight goal gave Sheffield United a 1-1 draw at Burnley as both teams' hopes of qualifying for the Europa League were dented.
James Tarkowski put Burnley ahead just before half-time, but Irish defender Egan netted with 10 minutes left.
United remain in eighth place, which could be good enough for a place in next season's Europa League should Manchester City's ban from European competition be upheld on appeal, while Burnley move up to ninth, two points adrift of the Blades.