Alexander Isak scored shortly before the hour mark to settle a difficult Premier League contest and give Liverpool another important three points.
Bournemouth had been unbeaten in their previous seven matches across all competitions before Liverpool ended that run with victory on Sunday.
Alisson admitted Liverpool were not at their best but praised the way they adapted to the challenge.
"Really important (win)," Alisson told Liverpoolfc.com after the match.
"I think every Premier League match away from home is tough to play. You have to fight a lot and this afternoon wasn't different for us.
"It wasn't the cleanest performance I think we had, but those games, if you don't play so well and then you still win it, keep the clean sheet, I think this brings confidence for the team for the rest of the season as well."
The Brazilian goalkeeper played his part by producing an important first-half save to deny Evanilson and preserve Liverpool's clean sheet.
It was Alisson's third consecutive Premier League clean sheet, and he said goalkeepers at top clubs must be prepared to make crucial saves even when they have little to do for long periods.
"At big clubs you're not going to have much to do sometimes and then you need to be ready to make one save that can help the team," he said.
"That's part of my job, I'm there for it. I'm happy that this afternoon it contributed to the three points and to getting the clean sheet."
Alisson also credited Liverpool's defenders for standing firm as Bournemouth pushed for an equaliser.
"Well done from the defenders as well, the team showed a really good fight," he added.
The goalkeeper also felt Liverpool improved after the interval and showed the flexibility needed to deal with Bournemouth's threat.
"Again, not the cleanest performance, we have to improve a lot on the ball, sometimes without the ball as well," Alisson said.
"But we improved in the second half, that's what you need to do as well in games: adapt to the opponent, to what the opponent is doing, to the difficulties you have in the match, and then fight against it and to get the result."
Liverpool will now turn their attention to a major test after the international break when they face Manchester City at Anfield on October 11.
