Saturday’s win against Arsenal ended a run of four straight defeats and followed victories at Manchester City and Tottenham earlier in the season.
City also pushed Liverpool and Manchester United hard in recent weeks, performances which made home defeats to Aston Villa, Leicester and Burnley seem all the more incongruous.
A trip to Crystal Palace on Saturday is next, and Begovic told Sports News HQ: “When it’s a big occasion, when a big team comes to town and you get a good result it gives everyone a boost, and hopefully now we can take that into the next games.
“We’re searching for that little bit of consistency where we can play against every team the way we have in those big games, so that’s something we have to work on.”
Begovic continues to be linked with some of England’s biggest clubs, but he said: “It’s just part of the game, you learn that as you get older.
“I’m just enjoying playing here and trying to get better all the time. It’s flattering to see yourself linked with certain clubs, but at the moment I’m just really focused on what I’m trying to do here.”