It came from Cody Gakpo's run and delivery down the right, with Isak converting from inside the area for his fourth league goal in five games.
Until then Bournemouth had been the sharper side. Evanilson's near-post flick from Marcus Tavernier's cross was turned away by Alisson Becker, but they remain winless in the league under Marco Rose, with three points from five games.
Rose told BBC Sport:
“It is disappointing to lose this kind of game when it is really tight. But we defended one situation not well enough. We didn't use our chances today and, in the end, we lose the game with one situation.
"We have to be sharper in both boxes.
“You don't get points for playing well, that is what we see. The fact is we have three points after five games when we could win every game. We have to speak about it, we don't have to change things in general - we just have to change details but we need to change those quick.
“We played a good game against Liverpool, who has some days more to recover. We looked fit, even at the end we were the team that went for a goal.
“It's about details and right now it is really disappointing, but I am quite confident we can change the situation points-wise.”
Asked whether it was a deserved win, Iraola said in his post-match press conference, published on Liverpool's official website:
“I think it was. I was disappointed with the first 20, 25 minutes, I think we didn't start the game with the intensity the game required. To win here is very difficult, I know it very well. Normally Bournemouth start really strong and it took us these 20, 25 minutes until we matched a little bit this level and from there and I think at the end of the first half and second half we played better, we had better situations with the ball and we could win the game.”
On Isak, he added:
“The striker's job is like this. He had a couple of crosses that he doesn't hit properly, one chance [he couldn't] control and this. I think he has to help us to be closer to the box, to play in the opposition half, to have meaningful possession and then, if the team is playing there, he will receive the rewards in terms of goals.”
Liverpool go into the international break sixth with nine points, and their first match back is at home to Manchester City on 11 October.