Liverpool were caught napping in the opening five minutes of the match, as Bournemouth had a goal disallowed for offside before taking the lead through Antoine Semenyo in the third minute.
However, the hosts did well to overcome their stutter, with goals from Mohamed Salah and Diogo Jota sealing the points after Luis Diaz had cancelled out Semenyo's strike.
"Horrible start to the game. I thought the first goal we conceded was offside and a wake-up call.
"OK, you take that. And then a minute later we conceded a regular goal which is not cool," Klopp told Sky Sports.
"We needed a few minutes to shake it off and then we were really good.
"The second half it changed completely because all of a sudden we were 10 men but scored the third goal, which is very important in this moment.
"It gave us energy. It is second match day and tough. It was surprisingly hot today, so it was intense for the boys but we fought through that and didn't concede anymore, got the three points and that is all that I need."
Klopp added that Liverpool had responded better to conceding a goal against Bournemouth than they did in their season-opening 1-1 draw with Chelsea.
"When we found our way into the game we played in areas we had to play and we were calm, which is good," Klopp said.
"I said after Chelsea, after we conceded I saw us getting into a rush. I did not see that today. We found our way back into it and played the game we wanted."
Liverpool's win came at a price, with Alexis Mac Allister being shown a red card for a high tackle in the 58th minute, meaning the Argentine midfielder will serve a three-match ban.
Mac Allister's sending-off seemed a harsh decision, with Bournemouth manager Andoni Iraola saying after the match that it "did not look like a red card".
Klopp agreed with his counterpart, saying: "I understand 100% how it looks for a ref in that moment but when you see it back, it is the inside of the foot and no power behind it.
"We saw harsher challenges which were not a red card and this shouldn't have been one. I don't know exactly what we have to do, but we should talk about it again."
