A stunning overhead-kick goal from Emre Can was the difference between the two sides as the Reds had to be patient to earn all three points.
Despite the blow, they managed to earn victory and the result boosts their hopes of finishing in the top four as they currently reside third in the table with three games left to play.
Manchester City sit three points adrift in fourth, while Manchester United are a further point behind in fifth, but both have a game in hand on Liverpool.
"It feels really good as you can imagine," Klopp told Sky Sports News after the game. "It was a difficult game to play, we started really well and then we had to change Phil, who is very important player, and Adam Lallana coming on after so long out.
"We started patiently, played football and tried to create moments. The goal was spectacular of course. It was clear we have to work and that is what we did. It won't be easy for anyone to win here.
"We needed a bit of luck but we worked hard for it. They did not have a lot of situations but it was difficult to defend. We needed to be careful and I think the boys did well. I'm fine with the performance."
Liverpool will face Southampton, West Ham United and Middlesbrough in their final games of the season.