Emery’s side were under the cosh in the first half at the Emirates Stadium, but Cech came to their rescue with a series of fine saves.
Having survived Everton’s opening salvo, the Gunners clinched their fourth successive Premier League victory thanks to second half goals from Alexandre Lacazette and Pierre-Emerick Aubameyang.
Lacazette’s sublime strike will stand the test of time, but Aubameyang’s was more fortunate as the Gabon forward was clearly offside.
While the predatory form of Arsenal’s strikers was encouraging for Emery, the Gunners boss was more heartened by his team’s first clean-sheet of the season.
It was a well-timed display from Cech after the former Chelsea star was replaced by pre-season signing Bernd Leno for Thursday’s Europa League win over Vorskla Poltava.
Emery’s decision sparked claims Cech was on the way out, but the Arsenal captain had other ideas as he rolled back the years with a vintage display.
“We are very happy with the goalkeepers. Petr has big experience and big quality. He has started the season very well,” Emery said.
“He is performing like we want and today he showed us he is very well.
“I’m surprised there are a lot of questions about Cech, because in the pre-season and in the first matches with us he played well.
“Leno is calm. He knows he is working well but Petr Cech is performing well.
“Leno will improve and learn with us. He has a big future.”
Cech, 36, added: “Everton created a lot of chances in the first half and we had to stick together.
“I’m delighted with the performance and the clean sheet. I had a lot to do today and I had to help the team stay in the game.
“As an experienced player, you have to do your part.”
– Heroics –
Cech’s heroics laid the foundation for Arsenal’s burst of two goals in three minutes, but it might have been different if Emery had been quicker with a planned substitution.
He was about to take off Lacazette just before the French striker notched his superb second goal of the season in the 56th minute.
“It is not important, maybe yes!,” Emery said with a grin when asked if he was set to substitute Lacazette.
Arsenal are up to sixth place in the Premier League, a welcome sight after the gloom that followed their two-game losing streak at the start of Emery’s reign.
However, Emery acknowledged there is still plenty of room for improvement.
“We want to control more. Above all in the first half, we give them the ball. We conceded two or three important chances,” he said.
“But we spoke at half-time about how to stop them and we had better positions when Everton attacked.
“Also we had to help each other and push more. I think we only conceded one chance after that.”
Everton’s second successive defeat left them without a win at Arsenal since 1996.
Their manager Marco Silva was frustrated with his side’s failure to take their chances and equally angry about Aubameyang’s offside goal.
“It’s a disappointing result. We deserved more. It didn’t reflect what happened on the pitch,” Silva said.
“We had three or four clear chances. It started in the first minute with a one v one against Petr Cech.
“It’s clear the second goal is offside. It’s really frustrating.
“It’s clear where the player who scored the goal is. He’s offside.”