Tom Lees put visiting Wednesday in the lead in the 36th minute, but the Cherries stepped up their game in the second half and went 2-1 ahead.
However, the Owls secured a point at Dean Court late on when midfielder Chris Maguire found the target from the penalty spot.
Lavery played the entire 90 minutes, and after full time, he posted the following message on Twitter.
Good result! That shows the fight that we have! Disappointed with my miss but I'll move on to the next one. Fans were top class again #swfc— Caolan Lavery (@clavery13) April 18, 2015
Over the course of the 90 minutes, Wednesday had just 38% of the possession, took 10 shots of which six were on target, and earned two corners.
In contrast, hosts Bournemouth had 62% of the possession, took 16 shots of which four were on target, and earned nine corners.
Wednesday are now 12th in the Championship table with 59 points from 44 matches.