A striker who represents the future for Everton and one who thought he belonged to their past gave Ronald Koeman the present he desperately needed against Sunderland. Dominic Calvert-Lewin and Oumar Niasse showed the Everton manager does have options up front after all as they halted the team’s sorry run of four games without a win or a goal.

Sunderland arrived on Merseyside with only one win from their previous seven matches, and their own dispiriting run continued as Everton rediscovered the cutting edge and solidity that had evaporated from their play in recent weeks. Koeman’s team selection was a key factor as he injected youthful energy, pace and better balance into his starting line-up. Confidence told too, growing through the ranks once Calvert-Lewin scored the first of his two goals before Niasse, 20 months after his £13.5m arrival from Lokomotiv Moscow, sealed victory with his first goal for the club.

“In a difficult period the best medicine is scoring goals and winning games,” said the Everton manager, rewarded for victory with a trip to Chelsea in the fourth round. “It is normal that you play with confidence at 2-0 but in a difficult game, and from the start, you have to show it too. Everyone worked hard and the final result was good. Let’s hope we have started to win at home and now we have three more [Goodison Park games] to go. If we win them all then I can go into the next international break with a smile on my face.”

Koeman’s changes included an overdue debut for the solid Jonjoe Kenny at right-back, a first home start for the recent signing Nikola Vlasic plus starting roles for Tom Davies and Calvert-Lewin. The four all impressed, particularly the 20-year-old striker who was denied his first Everton hat-trick when a diving header crashed against a Sunderland post.

Wayne Rooney, fined two week’s wages by Everton after his drink-driving conviction, money that will go directly to the club’s charity work in the local community, had asked to play but watched from an executive box instead. “It was not about the situation in court,” said Koeman. “I decided to rest some players and he was one. He asked me to play and I said no – Saturday.” A return against Bournemouth should not be so straightforward.

Sunderland should have capitalised on Everton’s early anxiety with a goal from James Vaughan, one of three former Evertonians in Simon Grayson’s starting line-up, but having dispossessed Muhamed Besic he shot wastefully wide.

The visitors would not get a clearer chance until they were two goals down.

Calvert-Lewin’s pace and touch gave him the edge over defenders Lamine Koné and Adam Matthews throughout and he brought a tedious first half to life shortly before half-time. Vlasic and Davy Klaassen combined for the England Under-20 World Cup-winner to steer a low finish beyond Jason Steele from 12 yards. The relief of a first goal in five matches had a noticeable effect on Everton’s confidence and performance levels.

Sandro Ramírez, stationed wide on the left, released Calvert-Lewin behind the Sunderland defence for a well-taken second. Once again the striker’s speed took him away from his marker and he met Sandro’s pass with a first-time, left-foot shot into Steele’s top corner.

Vaughan twice went close to a consolation for Sunderland with powerful headers from crosses by Donald Love and the substitute Callum McManaman but it was Niasse who struck the final blow. Everton’s forgotten man, who was told to find a new club by Koeman last summer and who almost joined Crystal Palace on deadline day this time, took Davies’s delivery expertly on his chest and stroked a cool finish into the roof of the net.