Goals from Nacho Monreal, Danny Welbeck and an own goal form Issa Diop sealed victory for the 'Gunners', despite Marko Arnautovic giving the 'Hammers' the lead in the first half.
The win was a much-needed one for Arsenal and new boss Unai Emery, who had fallen to back-to-back defeat to Manchester City and Chelsea prior to Saturday's game.
"I wouldn't describe it as relief," Bellerin told 'Arsenal Media' following the game. "We've been working hard now for the last two months, so we knew it was just a question of time.
"We had two tough games to start after lots of changes at the club. We knew it was a question of time, a question of scoring the chances that we had."
However, the Spanish full-back refused to get carried away, insisting that there is still plenty for his side to improve upon.
"We still could have been even more efficient," he continued.
"But even though we started behind in the scores, we turned it around and I was very happy with the reaction of the team as well.
"We knew we were playing at home with our crowd, it was a tough team, they've had a very good summer window.
"They've signed loads of top players, so even though they were still getting used to each other as well, we knew it was going to be tough. I think the score doesn't show how hard it was on the pitch."