City were heading towards a damaging draw before Nico O’Reilly’s 84th-minute strike put them on course for a vital victory that was sealed by Mateo Kovacic’s goal in the final seconds.
Pep Guardiola’s side climb to fourth place, but still just one point above sixth-placed Villa, who they face on Tuesday.
At a stage of the season when they are usually embroiled in a battle to win the title, City — champions in six of the previous seven campaigns — have been reduced to scrapping for a Champions League berth after a dismal season.
They remain on course to achieve their bare minimum target of finishing inside the top five after a hard-fought success on Merseyside.
In the 84th minute, Matheus Nunes’s low cross into the six-yard box was met by the 20-year-old O’Reilly, who got in front of Michael Keane to poke home from close range.
Kovacic wrapped up City’s victory on their last visit to Goodison Park before Everton move to a new stadium next season.
“To win here at Goodison Park, where Liverpool and Arsenal couldn’t, is massively important,” Guardiola said.
“Now it is in our hands. Three games at home, two away and hopefully we can achieve this big success to qualify for the Champions League.”
Villa travel to the Etihad full of confidence after a fifth consecutive Premier League win.
Unai Emery’s men bounced back from their Champions League quarter-final elimination to Paris Saint-Germain by ending Newcastle’s six-game winning run in style.
Ollie Watkins took less than a minute to justify Emery’s decision to recall him in the place of Marcus Rashford with a deflected opener.
Fabian Schar hit back to equalise for Newcastle, who were without manager Eddie Howe for the third consecutive game to pneumonia.
Watkins teed up Ian Maatsen to coolly restore Villa’s lead and kick off a three-goal blitz in 11 minutes just after the hour mark.
Dan Burn turned Jacob Ramsey’s cross into his own net before Amadou Onana fired into the top corner.
Villa’s victory means they close to within just two points of third-placed Newcastle with five games to go.
Already relegated Southampton’s last-gasp equaliser at West Ham ensured they will not have sole possession of the lowest points total in Premier League history.
Jarrod Bowen struck for the Hammers in the 47th minute, holding off Ryan Manning before lashing a fierce finish past Aaron Ramsdale.
But Lesley Ugochukwu equalised three minutes into second-half stoppage time as he lashed past Alphonse Areola.
Bottom of the table Southampton are now on 11 points — the same number Derby managed in 2007-08 — and have five games left to leave the Rams with the lowest total ever recorded in the Premier League.
Bryan Mbeumo scored twice as Brentford beat 10-man Brighton 4-2 to damage the Seagulls’ prospects of qualifying for Europe with their first home win since December.
Crystal Palace held on for a 0-0 draw against Bournemouth despite playing half the match with 10 men after Chris Richards’ first-half red card. (AFP)
