Theo Walcott netted the lone goal shortly after half-time as the hosts' only shot on target proved enough to end a three-game winless run.
They are now eighth in the Premier League – one point ahead of Leicester City – but the manner of a fifth successive victory in this fixture will have done little to assuage Allardyce's critics.
However, the 63-year-old was satisfied with an eighth Premier League victory in his time in charge and instead accused former employers Newcastle of negative tactics.
"What was wrong with our style today? We completely dominated the game," Allardyce said on Sky Sports.
"We were patient, we got the winner. You can't knock our football. You can knock some of the passing that goes astray buy you can't knock me for that, I don't pass the ball.
"When does every fan like the manager? I will keep doing what I'm doing because we keep winning and keep getting better.
"I think eighth in the league compared to fifth from bottom tells you we're getting better."
5 - Everton have won five consecutive top-flight games against Newcastle for the first time since January 1920. Edged.— OptaJoe (@OptaJoe) April 23, 2018
Newcastle, meanwhile, were left with a first defeat in five league fixtures as they missed a chance to move level on points with ninth-place Leicester City.
Magpies boss Rafael Benitez, who oversaw his 100th match in charge of the club, disagreed with Allardyce's assessment of Everton's victory.
"We played against a good team. The style and type of football wasn't amazing but they won and that's it," Benitez said.
"We had two or three chances. The final pass is the difference, especially in the Premier League. You have to take these chances to make a difference.
"I'm pleased with the effort of the players. To stay in the Premier League was the target and we are really clear with that."