The game was Sadio Mane's last for Liverpool before he heads off to the African Cup of Nations in Gabon and the Senegal forward was both hero and villain in an entertaining game.
Mane had looked to have given Liverpool all three points with a close-range finish following a second-half corner, but then gave away a penalty -- he was penalized for handball -- allowing Jermain Defoe to score his second penalty of the game.
The draw left Liverpool five points behind Chelsea, who play away at fifth-placed Tottenham Hotspur on Wednesday at White Hart Lane.
Manchester City beat Burnley 2-1 despite playing nearly an hour with 10 men after Fernandinho was red carded. City's win leaves Pep Guardiola's side seven points behind Chelsea.
City rivals Manchester United also stay in touch with the leaders after beating West Ham United at the London Stadium Monday evening with second-half goals from Juan Mata and Zlatan Irahimovic. West Ham played most of the game with 10 men after Sofiane Feghouli was sent off for a challenge on Phil Jones. Manchester United, in sixth, is now four points behind Liverpool in second.
Landmark game for Klopp
This was Liverpool manager Jurgen Klopp's 50th Premier League game in charge and Monday's result means the German has a record of 26 wins, 14 draws and 10 defeats.
Prior to Monday's match, Klopp had spoken of his unhappiness at the scheduling of this game, which came less than 48 hours after Liverpool's 2-1 win against Manchester City on Saturday.
"We can play better football but I'm not sure if you can play better with that break," Klopp pointedly told BBC Sport. "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 and not tell anyone, but nobody came and that was a good thing."
Since arriving during the 2015/2016 season, the former Borussia Dortmund coach has transformed Liverpool, buying cannily and improving the performances of a number of players already at the club.
Daniel Sturridge has struggled to earn a starting place under Klopp -- this was the first time the England international had begun a game since October 17 -- but the Liverpool striker provided the game's opening goal, cleverly arcing a header over Sunderland goalkeeper Vito Mannone.
Mannone had better luck dealing with a series of Sturridge shots with the Italian doing well to stop Liverpool extending their lead.
Sturridge, who has suffered a number of injuries over the last couple of seasons, had to be substituted just before Sunderland's second goal after coming off the worst in a challenge with Papy Djilobodji.