However, the hosts were behind inside nine minutes when Leon Osman struck from close range.
Walcott drilled home an angled effort in stoppage time to give the bruised Gunners renewed confidence for more stern challenges over the next seven months.
"We showed personality and mental strength - and that will be important for the rest of the season," said Wenger.
"I was not nervous at half-time because I had confidence and belief we would change the game.
"Everton had a good start, but in the last 20 minutes of the first half we were on top. Theo came on and did very well, he had impact in the game, but the whole team played very well and it was a good test for the team today."
Wenger added: "We played our season today. We could not afford to drop points and the way we responded was vital."
Everton defender Tony Hibbert and Arsenal full-back Gael Clichy were involved in a second-half scuffle following a late challenge on Denilson, which resulted in both players being cautioned by referee Peter Walton.
Wenger observed: "It was not a good challenge, but I do not think the challenge deserved a red card."
"That was not the normal Middlesbrough, who are a good team and it is always a difficult place to go."