Champions Liverpool unexpectedly dropped their first points at Anfield this season despite having 23 shots at goal.
Jurgen Klopp’s side looked set to maintain their perfect home record when Andy Robertson headed them in front in the first half.
But inspired Burnley goalkeeper Pope made a string of saves to keep the visitors in the game before Jay Rodriguez levelled with a neat finish immediately after the second-half drinks break.
MATCH FACTS
Liverpool (4-3-3): Alisson 6; Williams 6 (Alexander-Arnold 69 6.