Klopp’s side lead the Premier League table heading into the international break after moving two points clear of Manchester City, who have played one fewer fixture, but having come so close to a damaging draw at Craven Cottage.
Victory was ultimately secured in the 81st minute when Mane was perhaps fortunate to have been awarded a penalty following a challenge by the hosts’ goalkeeper Sergio Rico, and Milner scored from the spot.
“We want to stay in the race. That’s what we did,” said Liverpool boss Klopp. “Nothing is decided, but we can make sure we stay in the race.
“[First is] the best position you can be in. City don’t look as if they drop a lot of points, so we have to win, to win and to win. Now they have a game in hand but it was very important we won. We were really all very happy. You could see in our reaction.
“I didn’t see us panic. I liked how we reacted. The penalty doesn’t say we are sensational. Five or 10 minutes before the equaliser, we had so many little mistakes.
“We controlled the game completely, and then we gave Fulham the ball and the crowd woke up. But we had a very intense week, going to Munich, coming back, travelling to London, so I’m completely fine. Average game, sensational result.”
Klopp was asked whether Jordan Henderson would be fit for England’s Euro 2020 qualifiers with the Czech Republic and Montenegro, given his absence on Sunday with an ankle injury, and he responded: “He will go to the national team. He has two or three days still at Melwood, but it still looks positive.
“When it looks positive and he can train, he will go to the national team. He was not able to play, but it’s not that serious. In three or four days, it should be fine.”
Fulham, who are 13 points from safety with seven games to play, have lost their last seven games and their next fixture is at home to reigning champions City after the international beak.
Caretaker manager Scott Parker has so far overseen three defeats from three despite some improved performances and he is aware he needs to start producing results.
“I’m really disappointed,” the 38-year-old said. “We were well in the game. Then Sergio’s made a mistake and we couldn’t hold on to what would have been a massive point.
“I understand that, in the position I’m in, my main focus is seeing improvements. But I understand it’s about results. I want them more than anyone. Do I see an improvement? For sure I do. But we’re just trying to get some results.”