Leon Clarke outstripped defender Steve Cook to reach a long ball out of defence and then converted his 20th goal of the season with a left-footed shot.
Josh McQuoid and substitute Ryan Fraser both went close to equalising for Bournemouth before Carl Baker`s spot-kick earned the Sky Blues the points.
Bournemouth forced 17 corners to Coventry's three, but for all their pressure visiting goalkeeper Joe Murphy was called on to make only one top save from Matt Tubbs' volley.
"I thought we were very good," he said. "We had the majority of pressure, worked hard, but the ball did not fall for us in front of goal.
"We had some great chances early on and on another day we might have got five or six. It just did not happen for us.
"I would have been more unhappy if we had not created the chances, but we did. We are going through one of those spells and we have to accept it.
"I'm sure people will write us off, but we firmly believe we have a good squad and a good chance. Results can be deceiving. We have played a lot worse this season and won."