And with the Premier League race for the top four set to be more competitive than ever, Emery could target a fourth Europa League win in order to qualify for next season's Champions League.
Emery made eight changes for his opening Europa League game in charge - a 4-2 win over Vorskla Poltava on Thursday night.
After Alexandre Lacazette scored a fine opener at The Emirates on Super Sunday, Pierre-Emerick Aubameyang's goal moments later should have been disallowed for an offside.
But Emery was furious that Arsenal switched off in the closing stages to allow minnows Vorskla to score twice.
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.
Arsenal goalkeeper Bernd Leno has spoken for the first time about his frustrations at spending most of the season on the bench.
"It is a disappointing result for us".
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.More news: First iPhone XS, XS Max drop and liquid tests hit the web
Sarri has also vastly improved the performances of the players he inherited.
"I think the first 60 minutes were good, with us imposing our ideas against them and also, little by little in the match, winning battles on the pitch to be near to scoring", said the former Sevilla boss, who celebrated his 32nd win in the Europa League.
Former Arsenal man Theo Walcott was also guilty of squandering a good chance for the visitors, failing to get enough on his attempted chip over Cech when racing in behind, and Everton were so almost punished in the closing stages of the first half. "I am going to choose the best first XI [for every game]".
'But I think he did very well without the risk of losing another'. He showed us that he's very good and continuing to work hard with this spirit every day. "I think we only conceded one chance after that". Their side had won their previous three Premier League matches, and didn't look like stopping.
Everton are on their travels today as they head to the Emirates Stadium to take on Arsenal, with Marco Silva desperately looking for a response to last week's 3-1 defeat to West Ham United.
"It's something we analysed during the week".
Aubameyang said: "I just heard that I was offside like last season against Everton, but I have to say that it's a great job from Mesut and Aaron".