The Reds led for most of the game after Mane put them into the lead midway through the first half.
Mane's opening goal in the encounter, a stylish finish powered in after a one-two with Roberto Firmino, was the forward's 11th strike in his last 11 games and 20th overall this season.
Klopp laughed off his midfielder's impact after coming on as sub, but made more reference to the mindset of his squad and why he was convinced Liverpool would still win.
But they almost suffered a very costly European hangover against a Fulham side still 13 points adrift of safety.
For approximately seven second-half minutes, those defensive errors and Babel's subsequent equaliser for Fulham looked to have driven a stake into the heart of Liverpool's title challenge.
However, as a torrential rain shower swept over the banks of the Thames to make playing conditions hazardous, Liverpool's slender lead proved a unsafe one.
Liverpool centre-back Virgil van Dijk: "I am not giving excuses".
Disaster seemed to have struck for Liverpool when Van Dijk's header back towards Alisson didn't have enough purchase and Babel stepped in with the break of the ball to tap into an empty net. I love it here, there are lots of opportunities to develop an already really good football team.
Yeah, it's Hendo and Hendo will go to the national team.
"Listen, I said it two weeks ago, I said it after the last games, my message is the same, and to the players, this group have quality".
Fulham's recent results against Liverpool do not make for impressive reading.
Parker was easy on his goalkeeper Rico for the late penalty, which came after the goalkeeper had spilled a simple Mo Salah shot before grabbing at Mane.