Stoke shocked Premier League champions Manchester City in a 1-0 victory at the Etihad Stadium before the international break.
However, Mark Hughes' were beaten 1-0 by Leicester City on Saturday and have now lost both of their home matches in the Premier League so far this season.
Begovic has put those disappointments down to the strength of the top tier and is confident Stoke can build on last season's ninth-placed finish - their best in the Premier League.
"The Premier League now undoubtedly is the most competitive league in the whole world and anyone can beat anyone on any given day," the goalkeeper said at a gala dinner for his foundation.
"You have to be on top of your game, you need a bit of luck on the day as well. I think we performed pretty well for most parts of the game and we just couldn't score.
"Leicester defended really well, had a good game plan, caught us on the counter-attack and made the most of their one chance.
"It was one of those difficult days that happen in football but we move forward and if we keep putting the right effort into the game we'll get the right results.
"We've got a good squad, a great manager and I think we can achieve a lot of good things. If we can establish ourselves in the top 10 then that would be fantastic, and a really big achievement."
