To the list of times and weather conditions under which foreign pros should be tested at Stoke add “Saturday afternoon” and “blizzard”. Everton’s Cenk Tosun met the challenge here, claiming a second-half brace to earn victory for the visitors in a tempest of snow.

Stoke’s prospects of avoiding relegation worsened with the weather and they will feel aggrieved at this defeat. The Potters had Charlie Adam sent off in the first half during treacherous conditions. They also fought their way back into the match from one down, with Tosun’s opener itself prompting suspicions of offside. But a diving header from the Turkish international in the 84th minute proved the game’s decisive action.

Adam was sent off for the fourth time in the Premier League after sliding in on Wayne Rooney on the half-hour. The contact was not great but the tackle, especially given the surface, looked reckless. Paul Lambert accepted the decision after the match but railed against the lack of an intervention in Everton’s opener.

“I can understand why Martin Atkinson has given it‚” he said. “If you’re going in with studs in the modern game, you put yourself in a position. So I can understand that. But their first goal is offside. The linesman’s got to see that. How he’s missed that I don’t know.”

For Sam Allardyce there was no doubt about the red card: “It looked dangerous‚” he said. But with Everton recording their first away victory in the league for three months, Allardyce was happy to concentrate on more positive aspects of the match, such as the performance of Tosun, his £27m January recruit from Beşiktaş.

“It’s because of the quality of our frontman that we’ve won the game”, Allardyce said. “He’s adapting much quicker than lots of front men who come to the Premier League from abroad. He’s got the whole array of finishing in his locker; right foot, left foot, head. It also looks like he’s got the knack of being in the right place at the right time, which you can’t coach.”

Stoke looked as if they had stabilised after losing Adam, and Everton as if the match was passing them by until Tosun scored his third goal in as many games in the 69th minute. He looked offside in the first movement on a Rooney set-piece, his header then turned away by Jack Butland. Dominic Calvert-Lewin drove the ball back in again, however, and when Butland parried once more, Tosun forced home the rebound.

Lambert responded by going on the attack, bringing on Saido Berahino alongside Peter Crouch and replacing Ramadan Sobhi with Eric Maxim Choupo-Moting. The changes paid off quickly as Stoke equalised from their own set piece, Joe Allen swinging the ball to the back post, where Choupo-Moting bundled home.

Choupo-Moting was immediately forced off with a groin injury which he sustained as he scored, and five minutes later Everton won the match. Theo Walcott had time to shape a cross to the penalty spot and Tosun buried it with a diving header.

Stoke face Arsenal and Tottenham next, with a gap of three points between themselves and safety. “The effort and commitment was spot on from the players‚” Lambert said. “But it was a game that we lost and they’re certainly down in the dressing room.”