Spurs striker Harry Kane opened the scoring with his second league goal of the season, but Liverpool hit back through forward Diogo Jota and defender Andy Robertson. Spurs forward Son Heung-Min netted to level the scores again before Robertson saw red for kicking through the back of opposing defender Emerson Royal.
Spurs started brightly, and Kane took full advantage, slotting home a threaded through ball from midfielder Tanguy Ndombele.
Antonio Conte's side remained dominant as Son had two chances to extend Spurs' lead - first, with a lunging shot following a layoff from Kane; then, baring down one-on-one with goalkeeper Alisson -- but he squandered both. Midfielder Dele Alli missed another shortly after, unable to find the bottom right-hand corner after picking the ball up unmarked at the penalty spot.
Kane seemed to be in trouble of being sent off shortly after when he caught the boot of Robertson in a high-speed tackle, but he only saw yellow.
Klopp was also yellow-carded for his rant at the fourth official, sarcastically bowing as he went into the book.
"I was a bit more emotional in the game. He comes over and gives me a yellow card but I'd have preferred the right decisions on the pitch," the Liverpool boss said after the match.
Liverpool equalised on 35 minutes through Jota, who headed home a floated cross from Robertson.
Spurs were denied a claim for a penalty in the second half when midfielder Alli went down inside the box, and Liverpool raced to the other end of the pitch, with defender Trent Alexander-Arnold firing a cross that was headed in by Robertson.
Son struck back on 74 minutes as he slotted into an empty net after rounding goalkeeper Alisson.
Robertson saw red moments after the restart when he kicked through the back of Emerson. He was originally given a yellow, but referee Paul Tierney changed his mind after the Video Assistant Referee instructed him to review the pitchside monitor.
"I think it was an exciting game and we created many chances to score and win the game," Conte, whose side were back in action for the first time in two weeks after COVID-19 postponements, said.
"We have to be a bit disappointed for the draw. For sure we played against a really strong team.
"For the chances we created, we go home with the thought instead of one point it could be three points."
"We do not have the players. We could maybe have another [COVID-19] case or two. We have to think about it, we cannot just push it all through. We would prefer to play but we need some help. If we carry on we cannot carry on as usual."
Information from Reuters contributed to this report.