The victory made it six games unbeaten for the Glovers.
Carlisle's first-half performance was lacklustre and their best efforts both came from on-loan Sunderland midfielder Liam Noble, who tested goalkeeper Stephen Henderson with an early 30-yard shot then fired just wide after Carlisle had taken a quick free-kick.
Yeovil's Shaun MacDonald was a constant threat with his surging runs from midfield and he set up Obika for the opening goal in the 26th minute, the big striker curling his shot well out of goalkeeper Adam Collin's reach and inside the left-hand post.
Next it was substitute Andrew Tutte's turn to run at the defence. This time he chose to shoot and when Collin finger-tipped the ball away, Obika lashed the rebound into the side netting.
Yeovil's pace on the break led to their second goal in the 69th minute when the action switched swiftly from one penalty area to the other.
Carlisle's Peter Murphy had penalty appeals turned down as he went crashing in the Yeovil box, but the Glovers came away through Obika and Andrew Welsh, whose supporting run enabled him to find Bowditch with a low ball across the six-yard box, which left the striker with a simple tap-in.