The England midfielder struck first for Liverpool in the second minute of the game before doubling the visitors’ advantage with a sumptuous goal in the 57th minute.
Oxlade-Chamberlain’s goals were his first for Liverpool since his return to action after missing almost all of last season with a serious knee injury.
Klopp praised Oxlade-Chamberlain for his performance in what was his first Champions League appearance for 18 months but felt both the player and the team could have done better in their win against the Belgians.
'It was absolutely a great moment for him, but I would say with Ox his performance was exactly the same as the performance of the team - the goals were great but all the rest could have been better,' said Klopp after Liverpool’s 4-1 win against Genk.
'That's how it is, but it's no problem. The (second) goal was sensational and very important for us. Wonderful.
'On Sunday, Adam (Lallana) scored the goal, and now Ox has scored two goals. It's really nice. Absolutely great, a great story.'
Liverpool were lucky to go into the break with a 1-0 lead as the Reds made many mistakes but the hosts could not turn their chances into goals.
'We had good moments, a lot of good moments, and made a very good start' said Klopp.
'Then had another 10 or 15 minutes absolutely okay, exactly as we had to do it.’
'The plan worked okay then not anymore because we lost completely unexpected balls we were not ready for, which in the formation was not possible.’
'Sometimes we took in the wrong moments a risk, passing a one-two in a very tight situation makes not too much sense.
'We had to defend their balls with legs and had to run for it. They got a bit of momentum which after our start was completely unnecessary but that's what happened.
'It was a pretty stiff first half. Then we scored the second goal, by the way the first goal was very brilliant and the second goal even nicer, and all our goals were unbelievably beautiful.
'After the second it was a proper knock for Genk it looked like and we scored the third and fourth.
'We conceded one which wasn't cool but it's not the biggest problem in the world. We won the game, job done. That's it.'
