With the hosts failing to build on their 14th-minute lead, Genk's Tanzanian striker Mbwana Samatta powerfully headed in the equaliser at the near post from a corner five minutes before half-time.
But having scored twice in Belgium when Genk were beaten 4-1 on 23 October, Alex Oxlade-Chamberlain was the match-winner early in the second half, spinning and converting a low strike into the bottom corner.
Klopp remarked afterwards, "We didn't finish our situations in the way we could have done. We had a lot of shots and some really big chances that we usually use. We didn't do that and it kept the game open. That gave them a lot of energy.
"It was a good goal from them. It kept them alive. In the end, if we only score one and they score one it's a lot of effort. Ox [Oxlade-Chamberlain] got that wonderful goal but again, we don't finish the situation. The players came through but you saw after a while it gets hard to get that rhythm and at the end of the game it was not there. We got a little bit of trouble but not really. It was job done.
"All the goals we have conceded have been completely different. These things we can summarise so it's not about defending but about being 100% spot on in these situations. We know that and we need to work on that.
"We were the clear favourites tonight. If we scored two or three in the first half nobody would have thought that couldn't have happened. But it went 1-1 and then it got tricky. But now we are all good. Tomorrow will be a nice recovery day for everybody and then prepare for Manchester City."