Second-bottom Latvia had midfielder Aleksandrs Fertovs sent off for a reckless tackle on Elvir Rahimic after 11 minutes after which it was a matter of time before Bosnia scored.
Winger Senad Lulic got his first international goal and set up a second for Bosnia as the Balkan nation moved to 16 points from six matches and extended their lead to six points over second-placed Greece who visit Lithuania later on Friday.
Having missed a hatful of chances in the first half after Fertovs was dismissed, Bosnia finally broke the deadlock in the 48th when Lulic turned in a Mensur Mujdza cross from the right.
Roared on by several thousand visiting fans, Bosnia doubled their lead five minutes later after Lulic curled in a delightful cross for striker Vedad Ibisevic to head home from 10 metres.
Midfielder Haris Medunjanin scored the third soon after the hour with a brilliant shot from 25 metres into the top right corner, leaving goalkeeper Andris Vanins rooted to his line.
Vanins had made a string of super saves to deny the visitors even more goals but was powerless to stop his side's mauling in the closing stages, as Bosnia continued to attack.
Midfielder Miralem Pjanic made it 4-0 with a crisp low shot from 12 metres in the 80th and Bosnia striker Edin Dzeko completed the rout barely a minute later with a cool finish after he raced clear of his marker.