First-half goals from Hope Akpan and Joe Walsh gave Crawley the early lead in the League One clash at the Broadfield Stadium on Tuesday night.
Lee Barnard gave the Cherries hope in the second period before Nicky Adams added Crawley's third.
Bournemouth now sit fifth-from-bottom in League One having won only one of 12 games this season but Groves, who took over the reins on a permanent basis in May, expressed his desire to turn around the club's fortunes.
Responding to being questioned on the pressure he was under, Groves said: "There is always pressure and you hurt.
"But you keep working. I'm not resigning. In football, you don't quit because you are doing the work.
"We have lost key people and it's important that people take responsibility on the pitch."
However, Groves was subsequently removed from his post at Dean Court on Wednesday.