The Reds appeared to have fallen behind at Anfield when Aleksandar Mitrovic found the back of the net, but his effort was ruled out by match officials required to make tough calls in the heat of the moment.
With Fulham feeling hard done by, their mood was further soured inside 14 seconds as Liverpool burst to the other end of the field and Egyptian frontman Salah broke the deadlock.
Xherdan Shaqiri would add a second for the hosts, helping them to preserve their unbeaten Premier League record in 2018-19, with Klopp happy to claim a slice of good fortune in an ongoing pursuit of title glory.
He said of the Mitrovic decision: “I have seen it once, I'm not sure. What did the guys who have seen it 20 times say? It's still difficult to tell?
“I don't want to compare it, but against Arsenal we scored a clearer goal which was not offside and didn't get it. You cannot change it.
“The most important thing is reacting in that situation that quickly, that was brilliant.
“In the end, we were a bit lucky. It's like it is. We cannot change it. Fulham cannot change it. The reaction was brilliant.”
Victory for Liverpool lifted them back to the top of the table, ahead of a Manchester derby date between City and United, and saw them match their best points return of 30 through the opening 12 games of a Premier League season.
Klopp continues to see plenty of positives from his troops, with unexpected obstacles overcome against Fulham to collect a hard-fought win.
“In a few moments it was tough, and in a few moments it was not that tough. When we saw their line-up we were a bit surprised – how could we prepare for that line-up?” added the German tactician.
“It was the first this season, I think, that Fulham have played like this, with the players they lined up today with four or five centre-halves.
“So far, Fulham have always had a football-playing idea. Today, they were winning second balls, had everyone on their toes, and while they played a bit of football there was a lot of long build-up, and Mitrovic did outstandingly well in those situations.
“That was a bit tricky, but apart from that, the boys from the beginning did really well.
“We created, we passed, we accelerated in the right moments. We had big chances early with Sadio [Mane], with Shaq, with Mo in the box, when they were blocking in the last second and the goalie could make a save.
“I don't know exactly how many chances we had today, but it wasn't important as we scored twice.
“Maybe it was a bit strange how we scored, one after a quick restart and one in the second ball phase at a set-piece.
“But it’s important you have these situations in your locker and you can use them. I'm completely happy, apart from scoring more goals. It was a really good performance.
“We had to work hard, we won it. All good.”