The 23-year-old failed to connect with the ball when he slid out to intercept Mark Noble's pass, allowing Demba Ba to run on to score the Hammers` 21st-minute opener.
However, Begovic made two superb saves in the second half to deny Carlton Cole before Thomas Hitzlsperger struck in the 83rd minute to complete Stoke's heaviest league defeat of the season.
"It was unfortunate," he told The Sentinel. "I remember the ball being played through and I came sliding in, but the ball bounced a bit weirdly and went over my foot. It looked a bit silly, and obviously it is my fault for making a challenge and not making it well. I will take that on the chin and move on I guess, and hopefully rectify it in the next games."
City are only four points from the Premier League relegation zone and six short of the likely 40-point safety mark with nine games still to play. However, Begovic believes no-one in the Stoke camp is taking survival for granted.
He added: "There are a lot of teams in it and we aren't safe by any means. We have to pick ourselves up and start picking up points fast to push away from the relegation zone and establish ourselves in the top half of the table."
Source: PA
Source: PA