The Reds left it late in the round of 16 first leg at the San Siro thanks to goals from substitute Roberto Firmino and Mohamed Salah.
The visitors rode their luck at times but Inter could not breach a defence which was well marshalled by Virgil Van Dijk.
Klopp acknowledges his side did not play to their usual standards but is delighted to have the healthy advantage ahead of the second leg at Anfield.
He told BT Sport; "We had to fight. We started really well, played around their formation in the beginning, but then the direct play with the switches to the wing-backs we didn't defend too well.
"Too often our full-backs were alone in the one-on-one situation, it doesn't work properly like that.
"In a game like this you have to keep the ball longer and that's what didn't happen long enough.
"But you cannot come here and hope that you have a brilliant day and that's the only chance to get a result.
"We had not a brilliant day but a good enough day to be a deserved win. We scored two wonderful goals, yes they had their moments [and] counter-attacks, but mainly counter-attacks when we lost the ball in the wrong spaces.
"It's another adult performance and I'm really happy about that."
