Aged four, Begović and his family emigrated from their war-torn home of Trebinje to Kirchhausen, a borough of Heilbronn, near Stuttgart in Germany. Begović developed a keen interest in football and played with the local youth club before the Begović family moved again this time to Edmonton, Canada. After playing with Minor League side Southwest Sting he earned a call up to the Canada U-17 squad. In the summer of 2003 he signed for English club Portsmouth after impressing on trial. After playing with Portsmouth's youth team for two years he moved on loan to Belgian side La Louvière in 2005.

After gaining a UK Visa in the summer of 2006 he was able to get match experience in the Football League playing for Macclesfield Town, Bournemouth, Yeovil Town and Ipswich Town. He finally made his Premier League debut for Portsmouth in May 2009 and took over from David James in 2009–10. However Portsmouth's financial problems led him to be sold to Stoke City in January 2010 for a fee of £3.25 million. He began as second choice to the veteran Thomas Sørensen before becoming a regular in the side in 2010–11 and 2011–12. In 2012–13 he played in every league match and his performances saw him win the Player of the Year award. In November 2013 Begović became the fifth goalkeeper to score a goal in the Premier League, scoring from 97.5 yards against Southampton after 13 seconds. He joined Chelsea in July 2015.

Begović represented Canada at youth level playing for them at the 2007 FIFA U-20 World Cup before deciding to change his allegiance to Bosnia and Herzegovina in 2009. He soon established himself as Bosnia's number one and helped them reach the 2014 FIFA World Cup.