Jay Rodriguez equalised for Burnley in the 69th minute after Andy Robertson had given Liverpool a first-half lead.
Liverpool, on 93 points, must win their last three games if they are to beat Manchester City’s record total of 100 points from two years ago. Burnley are ninth on 50 points.
The last time Liverpool failed to win at Anfield in the league was in January, 2019, when they drew 1-1 with Leicester City.
Burnley could even have pulled off a famous victory when Johann Berg Gudmundsson stuck the bar three minutes from fulltime.
Burnley keeper Nick Pope, continuing to make the case for promotion to the England number one spot, kept Liverpool at bay with a series of good saves, including a brilliant reaction to keep out a powerful close-range effort from Mohamed Salah in the 18th minute.
But Liverpool’s domination finally came to fruition in the 34th minute when Fabinho floated in a cross from the left and Andy Robertson met it with a perfectly angled header into the top corner. Pope was outstanding again to keep out a Sadio Mane drive with the Burnley keeper largely responsible for the Clarets heading in at the break only a goal down.
Liverpool’s Roberto Firmino should have had his first league goal at Anfield this season but scuffed his shot against the post in the 49th minute.
Burnley gradually grew into the game and drew level with a superb finish from Rodriguez after James Tarkowski had headed down a deeply hit free kick.
Liverpool had one more opportunity in stoppage time but Salah’s weak effort was easy for Pope.
“We should have scored two, three or four goals at least,” said Liverpool manager Juergen Klopp.
“The referee let a lots of challenges go so it was clear that if the ball comes into the box it was dangerous. They did what they are good at and I respect that,”
For moments it was Liverpool against Pope, he did really well, but we should have scored. We didn’t close the game and they took their moment. It feels like we lost a game,” he said.
Burnley boss Sean Dyche hailed the way his side approached the second half.
“Second half I reminded them that the pressure was off, just go and play. Get it forward and ask them more questions. You’ve got to find a moment, and we’ve been great at that,” he said.
“We hit the bar late on but I think three points would have been over-egging it, we defended well and our keeper played well,” he said.