Substitute Thompson was on hand at the near post to turn home a low cross from right-back Cliff Byrne to deny the Cherries what looked to be their first away win of 2012.
In a lacklustre first half, it was the visitors who were the better of two poor teams. Striker Lyle Taylor squandered a hat-trick of good chances, twice volleying over from the edge of the six-yard box and then firing just wide with an overhead kick.
Bournemouth were finally rewarded in the 42nd minute when former Scunthorpe loanee Donal McDermott forced his way into the penalty area and squeezed in a right-footed shot off the inside of the far post.
At the other end, Scunthorpe had a lot of the ball but offered little threat, midfielder Robbie Gibbons going closest when he fired a shot across the face of the goal and just wide of the far post.
It was a similar struggle in the second half for Scunthorpe before Thompson struck. But in the closing stages, it took a fine fingertip save from goalkeeper Sam Slocombe to deny substitute Zavon Hines a winner for Bournemouth.