Riyad Mahrez and Raheem Sterling calmed Manchester City’s nerves on Saturday as the Premier League champions overcame a searching test to beat feisty Everton 3-1 and keep leaders Liverpool in their sights.

Pep Guardiola’s team came into the match at Goodison Park knowing they could not afford to fall further behind Jurgen Klopp’s high-fliers, who earlier extended their perfect record with a win at Sheffield United to stretch their lead to eight points.

It appeared to be normal service for City when they took the lead midway through the first half through Gabriel Jesus but Everton hit back nine minutes later through Dominic Calvert-Lewin.

Both sides had chances after the break in an entertaining contest but Mahrez’s free-kick and Sterling’s late strike put the game beyond the home side and allowed City to close to within five points of Liverpool.

“We know if we have a setback in a match we stick with the gameplan, that is the most important thing. The chances will come,” said Sterling.

“Coming here is always one of the most difficult fixtures in a season. The pitch is narrow and they are always up for it but we got over the line.

“It is still early in the season and there are still (more than) 90 points to play for. We are behind at this moment in time but the league can change very quickly and we can stay focused.”

City, playing in a garish yellow-and-pink kit, took a deserved lead in the 24th minute when Mahrez found Kevin De Bruyne, who placed a delightful curling cross onto the head of Jesus, an unused substitute during City’s 8-0 demolition of Watford last week.

The home side responded brightly, with Gylfi Sigurdsson going close, and pulled level in the 33rd minute when Seamus Coleman chipped over goalkeeper Ederson from a tight angle and Calvert-Lewin headed in on the goal line.

City had chances to restore their lead before half-time but Jesus blazed over and Gundogan fired straight at Everton goalkeeper Jordan Pickford.

Early in the second half De Bruyne hit the side-netting before Ederson was called into action at the other end, tipping over Yerry Mina’s close-range header as Everton went toe to toe with last year’s domestic treble winners.

Sterling missed a glorious chance to put City back in front on the hour mark, side-footing wide from Mahrez’s incisive pass.

But the visitors eventually got the goal they craved when Mahrez curled a free-kick into the bottom corner past a despairing Pickford with about 20 minutes remaining.

Sterling made sure of the win when he smashed an effort against the underside of the bar and goalline technology confirmed it had crossed the line.