The Reds came into the game at Anfield buoyed by their sensational Champions League comeback against Barcelona in midweek and knowing they had to win while hoping Brighton would do them a favour against City.
And they stuck to their part of the bargain when Mane fired in the opening goal of the game just after the 15-minute mark.
He grabbed his second of the game nine minutes from time to move level on 22 goals for the season, along with team-mate Mohamed Salah and Arsenals Pierre Emerick Aubameyang at the top of the scoring charts.
However, Citys 4-1 mauling of Brighton meant Pep Guardiolas side finished a point ahead of the Reds, who unbelievably missed out on their first top-flight title since 1990 despite picking up 97 points.
Liverpool made two changes following the 4-0 win over Barcelona in midweek as two-goal hero Georginio Wijnaldum started in place of James Milner, while Salah was back in for Xherdan Shaqiri.
Wolves stayed unchanged after their 1-0 Premier League win over Fulham last Saturday.
Liverpool dominated from the first whistle as they pushed Wolves back into their own half for long periods.
It was no surprise when they eventually broke them down on 17 minutes as Trent Alexander-Arnold did well down the right and whipped a superb cross in for Mane to finish clinically from 10 yards.
It was almost 2-0 on 25 minutes when Andrew Robertson let fly with a fierce strike from 25 yards which Rui Patricio did well to push away to his left.
News of Brighton going ahead and then Citys rapid equaliser at the Amex Stadium appeared to disrupt the flow of Liverpools play, although Salah was not too far off the mark with a volley from the edge of the box on 33 minutes.
Wolves had a half-chance three minutes later when Willy Boly flicked the ball on towards Raul Jimenez but he could only scuff his shot wide of Alissons goal.
The atmosphere went flat as news of City taking the lead on the south coast filtered through and it was Wolves who started to get a grip on proceedings.
And they were inches away from an equaliser just before the interval when Diogo Jota moved the ball wide to Matt Doherty on the right edge of the area and he thumped a shot towards the far corner which crashed off Alissons crossbar.
Wolves made the brighter start to the second half with Jonny seeing a shot in the area blocked, before Jimenez slashed a strike wide on the hour mark when well placed.
Liverpool almost went two-up on 63 minutes when a quick counter-attack saw Mane set up Divock Origi in the area via a deflection off a defender and the strikers shot on the turn was brilliantly diverted wide by Conor Coady.
City going 3-1 ahead at Brighton effectively ended Liverpools title dream and Wolves went close to equalising when Jota looked to get on the end of Dohertys ball into the area only to be denied by Alisson who produced a sliding save.
Jota should have scored on 73 minutes when he was picked out by Jimenez in the area but sent his header from eight yards straight at Alisson.
Despite knowing City had grabbed a fourth, Liverpool managed to galvanise themselves in the closing stages and Mane grabbed his second of the afternoon when heading in another sublime Alexander-Arnold cross on 81 minutes.
Virgil van Dijk then watched his header bounce off the crossbar two minutes later from yet another Alexander-Arnold assist.