Ange Postecoglou suffered the intense frustration of watching his Tottenham side concede a late goal to draw 2-2 with rejuvenated Wolves.
Defending champions City travelled to the King Power drained of confidence after just one win in all competitions since late October.
Their frailties over the past two months, which resulted in nine defeats in their previous 13 games, were on display again.
But Savinho's first-half goal and an Erling Haaland header after the break earned a much-needed three points.
Leicester provided a stern test and might feel they could have got something out of the match as Facundo Buonanotte hit the post and Jamie Vardy missed a key chance at 1-0 down.
City's bid for a fifth straight title looks to be over as they are 11 points behind leaders Liverpool, who have played two games fewer.
But the result lifts them to fifth in the table and within striking distance of the top four.
Savinho opened the scoring midway through the first half with his first goal for City while Haaland secured the win in the 74th minute -- sparking wild celebrations among the City players.
"We needed it, all of us," City boss Pep Guardiola told Sky Sports. "It was not the ideal performance but hopefully the victories will give our mood a better position.
"It's important to win, it was tight, second half. We didn't have enough energy to sustain 90 minutes but hopefully in the New Year we can bounce back a bit from a bad moment."
Leicester remain rooted in the bottom three after losing their past four games under new manager Ruud van Nistelrooy.
Forest riding high
Nuno Espirito Santo's Forest beat Everton to win their fifth straight Premier League match, leapfrogging Chelsea and Arsenal.
Chris Wood chipped over Jordan Pickford for the opening goal in the 15th minute after being set up by Anthony Elanga.
Wood turned provider for Forest's second, which came just after the hour mark when Morgan Gibbs-White lashed home to make it 2-0.
They are five points behind Liverpool, who have played two games fewer.
Wood said Forest were not obsessed with their position in the league table but he admitted qualifying for European football would be a "dream".
"For the club and the fans it's the dream. It's the Premier League and it's the toughest league in the world," he told the BBC.
Spurs were on course for just their second Premier League win since late November against Wolves.
Hwang Hee-chan scored an early opener for the visitors but were ahead for just five minutes before Rodrigo Bentancur headed an equaliser.
Brennan Johnson powered Spurs into the lead in the dying minutes of the first half to put his side on track for all three points.
But Jorgen Strand Larssen produced a brilliant finish from the angle to earn Wolves a share of the points.
New Wolves manager Vitor Pereira remains unbeaten, with two wins and a draw in his first three matches in charge at Molineaux, taking the club out of the relegation zone.
Crystal Palace won just their second home league game of the season, seeing off bottom club Southampton 2-1, thanks to goals from Trevoh Chalobah and Eberechi Eze.
Dango Ouattara scored an 89th-minute leveller as Bournemouth drew 2-2 at Fulham.
Liverpool travel to West Ham for the late kick-off while Chelsea are not in action until Monday, with Arsenal travelling to Brentford on Wednesday.
