Everton's quest for Champions League qualification is back on track but only after a series of controversial detours against West Ham. Anthony Taylor's direction of proceedings baffled and infuriated both teams and caused swift shifts of momentum at crucial times.

Everton prevailed and even though their third victory in 12 Premier league matches lifted them into the top four before Chelsea's home game with Aston Villa on Sunday, David Moyes's joy was tempered by his dismay at the display of a referee who disallowed a goal and issued two contentious red cards.

Sam Allardyce was even more aghast and the West Ham manager will seek a meeting with Taylor's superiors to ensure the 34-year-old official never repeats this performance.

West Ham benefited from the referee's first significant intervention, which came in the 12th minute. Leighton Baines delivered a typically precise corner to Leon Osman, who nodded into the net from 10 yards. The midfielder's joy was curtailed by the referee, who, on the advice of his assistant, chalked off the goal. The grounds for that decision were far from obvious, the best guess being that Victor Anichebe was penalised for some gentle jostling with goalkeeper Jussi Jaaskelainen.

"I was really annoyed with that decision," Moyes said. "I can't see how the linesman can give that decision when he is 50 yards away and the referee is only 15 away. It was tough to take."

It was even tougher to take two minutes later, when the reprieved hosts went in front. Matt Taylor slipped a pass through to Carlton Cole, who shrugged off Johnny Heitinga before firing a low shot into the net from the edge of the box for only his second goal of the season. Heitinga struggled to contain Cole all afternoon and had only been assigned that task because Phil Jagielka was shunted to right-back in the absence of the injured Seamus Coleman.

West Ham, however, failed to create any further chances in the first period, and Everton had the better of the play but they too struggled to conjure clear opportunities.

Nikica Jelavic should have equalised in the 49th minute when the ball broke to him six yards from goal. The Croatian shot smartly on the spin but straight at Jussi Jaaskelainen. The mounting threat from Everton persuaded Allardyce to make a change, and the substitute Modibo Maïga shot weakly wide after a counterattack led by Cole.

Four minutes later, Everton drew level thanks to a superb finish by Victor Anichebe, who headed home a Steven Pienaar cross into the corner of the net from seven yards.

West Ham had barely digested that when another controversial decision by the referee left them spluttering. Taylor showed Cole a red card in the 67th minute after the forward challenged Baines for a bouncing ball. While Baines stooped to head it, Cole raised his foot to kick it and instead prodded the defender near the chest. Baines did not even pretend to be hurt and both managers confessed to being shocked at Cole's expulsion. Allardyce wondered how Cole's challenge differed from a similar incident in the first half involving Anichebe and James Collins.

"We didn't get consistency from the referee and that had a massive effect on the result," Allardyce said.

Everton took advantage of the swing in fortunes. In the 73rd minute, Osman dribbled past two defenders to the byline and pulled the ball back across the face of goal, where it ricocheted off Matt Taylor and then Pienaar before trickling into the net. Jelavic missed a wonderful chance to increase Everton's lead when he blasted the ball into the side netting from four yards in stoppage time, and Kevin Nolan shot wide when he only had Tim Howard to beat.

In between those two chances, referee Taylor found some consistency, dismissing Darron Gibson for a challenge on Mark Noble similar to Cole's one on Baines. "After the first one he probably felt that if there was another foot raised high he'd have to do the same," Moyes said. "I said to him, I don't think that's the way to referee."

Both managers confirmed they will appeal against the expulsions and Allardyce said he will take the matter further. "I'd like to know what the referees' coaches or whoever think about what happened and what is the right way forward for Anthony Taylor," he said. "I'll pursue that quite vigorously to see what answers they give me. He has a difficult job but if he's not getting the right feedback and coaching and just told 'Well done, carry on,' then we've got a real problem."