The Cherries opened the scoring just 30 minutes into the clash through Brazil international Evanilson, before the striker's international compatriot, Matheus Cunha, leveled for Wolves in the 60th minute.
The contest ended in a stalemate regulation, and extra time was forced as a result, but neither team found a winner, moving the match to penalty kicks. However, as the final whistle neared, Cunha was handed a straight red for a headbutting offense on Bournemouth defender Milos Kerkez in the 120th minute.
In kicks from the spot, Wolves missed their fifth and sixth attempts, with Matt Doherty shooting wide and Boubacar Traore hitting the post, to see the Cherries through.
Adams, meanwhile, played all 120 minutes of the contest, completing 62 passes and registering 95 touches on the ball. He did not attempt a penalty.