Mr Besigye was bundled into a police van when he tried to leave his home, where he had been under house arrest, witnesses said.
He has dismissed Thursday’s poll as “fraudulent”.
Mr Museveni won by a landslide to extend his 30-year rule.
In a BBC interview, he rejected allegations of vote rigging, and accused Mr Besigye of planning to incite violence.
Responding to criticism from European Union observers that the electoral commission “lacked independence and transparency”, Mr Museveni told the BBC’s Zuhura Yunus that “those Europeans are not serious”.
“Transparency is what we’ve been voting for,” he added.
Official results gave Mr Museveni nearly 61% of the votes, with Mr Besigye taking 35%.
Mr Museveni seized power in 1986 and is credited with restoring stability to Uganda. However, critics say he has become increasingly authoritarian.