When Rodrigo Bentancur pounced on Neto's careless pass from the back, it broke for Sarr, who sent it into the far corner to open the score.
Spurs, still smarting from their midweek 4-2 setback to Brighton, sat helplessly as Bournemouth had plenty of opportunities and were certain to tie the score.
Son Heung-min extended his team's lead with a fantastic left-footed shot that went in off a post, and Richarlison rushed home to seal the victory for Ange Postecoglou's team, but the visitors lacked a precise edge in the final third.
Even though Alex Scott scored a late consolation goal, the Cherries will probably be wondering how they let the score slip past them and end up empty-handed in London.
Dominic Solanke's three-game scoring streak came to a stop, as Andoni Iraola's team suffered their first loss in seven games.