Matt Ritchie had a chance to open the scoring for the Cherries inside the first minute of the match, but his half-volley was dragged just wide.
Newly-signed keeper Artur Boruc came under pressure on his Cherries debut as Matej Vydra tried to force the ball in from a Lloyd Dyer cross, before the striker set up Daniel Pudil for a volley later in the half.
Bournemouth thought that they had broken the deadlock at the start of the second half when Wilson tapped in from a rebound, but his finish was chalked off for offside.
Eddie Howe's men continued to create the better of the chances, with Junior Stanislas twice missing the target, and they finally managed to take the lead when Harry Arter converted a low drive from the edge of the box with just over an hour played.
Fernando Forestieri came off the bench as Watford tried to get a goal back, but the forward could only fire wide with a long-range effort.
The hosts did manage to get back on level terms inside the final 10 minutes, though, as Cathcart blasted the ball in from close range following a free kick.
Watford could have won it in the final moments when Odion Ighalo tried to smash home the winner after collecting a pass from Forestieri, but Boruc managed to make the save.