Cech has been challenged by Emery to use his feet more so the team can play the ball out from the back.
Sprinting onto Gylfi Sigurdsson's pass, Walcott had a clear sight of goal but shot straight at Cech in a familiar sight to Arsenal fans who endured the winger's erratic displays for so long.
"Petr, he has a lot of experience and quality", Emery said. "I think it's important".
Arsenal look far more unsafe with the Frenchman through the middle and Aubameyang coming in from the left to join him.
Cech made five stops in the first half that ranked from decent to excellent and, more than anybody, it was the 36-year-old that created the platform for victory.
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His link-up play and chemistry with Hazard make him a nuisance to defenders, for which Hazard is very grateful. Racing on to World Cup-winner Giroud's flick, Hazard scampered past Joe Bennett and drilled home the leveller.
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.
Arsenal's next opponents are Everton, another team that have struggled defensively so far, and they will be hoping Pierre-Emerick Aubameyang can take advantage of that weakness, especially after the striker rediscovered his scoring form with two goals against Vorskla.
Cazorla rejected the notion that Arsenal were in "a comfort zone" under former manager Wenger - who was replaced after 23 years in charge by Emery in May - but feels that they could have come closer to success in the Premier League.
Cech kept Arsenal in the game during the first half, with Marco Silva feeling a 2-0 reverse was an unjust scoreline given Everton's performance and replays clearly showing Aubameyang was offside when he scored. "Assistant referees can make them".
"It [VAR] will help for sure", Silva said. I didn't just come to learn from Petr.
The win - and Arsenal's first clean sheet of the season - pushes the Gunners up to sixth in the table, while the Toffees stay in 12th place.
Thursday's game saw summer signings Bernd Leno, Stephan Lichtsteiner and Lucas Torreira all make their first starts for the club, while Alex Iwobi and Henrikh Mkhitaryan - who were on the bench in the last couple of league games - also impressed Unai Emery.