The result left Liverpool on 44 points from 20 games, five behind leaders Chelsea, who play their game in hand at Tottenham Hotspur on Wednesday, and two clear of Manchester City.
Sunderland's Defoe cancelled out Daniel Sturridge's early opener in the first half and rifled in his second spot kick late on after Sadio Mane restored Liverpool's lead with a tap-in.
Individual mistakes cost Klopp's side as Mane handled a free kick for the second penalty after Ragnar Klavan gave away the first with a clumsy challenge on Didier Ndong.
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"I am not able to explain it because I don’t know exactly what I saw, my team were fighting but I wasn’t sure if they could do it," Klopp told the BBC.
"We can play better football but I’m not sure if you can play better with that break," he added, referring to a congested holiday fixture list which saw his team play twice in three days including Saturday's 1-0 win over Manchester City.
"I don’t know how it feels when you have to do the things you have to do today.
"I told the players if nobody wanted to play I would never speak about it and not tell anyone, but nobody came (forward) and that was a good thing. About the football we played, I actually have no idea how to speak about it."
Liverpool dominated the match and Sunderland only stayed afloat thanks to a string of superb saves by goalkeeper Vito Mannone, who kept out a barrage of shots either side of Sturridge's opportunist 19th minute header.
More poor defending following a corner allowed Liverpool to reclaim the lead in the 72nd, after Defoe had equalised the first time, as Mane was left with a simple finish, but the Senegal winger then turned villain with a costly error.
A free kick from the edge of the box struck his outstretched arm and Defoe was again clinical with his 84th minute spot kick to lift some of the pressure off manager David Moyes after the Black Cats were thrashed 4-1 at Burnley on Saturday.
"I’m really pleased after the few days we’ve had," said the Sunderland boss, whose third-bottom side stay in the relegation zone with 15 points from 20 games, a point off Crystal Palace.
"I thought we did quite well, matched Liverpool’s energy for long periods of the game and deserved a draw. We had big chances as well. I never thought we were out of it, the important thing was to not concede a third goal."
(Writing by Zoran Milosavljevic; Editing by Ken Ferris)
