The defeat at Craven Cottage on Saturday was Stoke's third in a row and saw them slip to 16th in the Premier League.
Darren Bent struck late on after leaving Begovic and Robert Huth on the floor inside the penalty area.
Mark Hughes has implemented a new style at the Britannia Stadium that looked to be having a big impact after a relatively strong start to the season but Begovic knows that they must do better when they return after the international break.
"We only had ourselves to blame for losing," the Bosnian told The Sentinel.
"If you have most of the possession but don't take your chances or make the most of your play, then the least you should come away with is a point.
"It's tough to take, but we have to get going. It's all part of the process. It's a long season and we have to get better.
"We have the quality in the team, we just need a couple of things to turn our way."
Begovic will line up for Bosnia as they attempt to reach the World Cup finals for the first time – they face Liechtenstein and Lithuania in Group G and victories in both will guarantee their qualification.