Ashley Barnes wrote his place in the Clarets' history books but left Anfield frustrated his goal was not enough to earn a deserved reward.
Barnes became the first Burnley player to score at Anfield since Peter Noble in 1975 as he gave Sean Dyche's men an early lead.
But a Liverpool equaliser with the last kick of the first half got the Reds back in the game before the top-four hopefuls sealed a 2-1 win with Emre an's 60th-minute strike.
"It was nice to get a goal, I haven't had one for a while, but disappointing to come away with the result that we have," said Barnes, whose fifth goal of the season was his first in open play since the end of December.
"To concede right on half-time was a kick in the teeth for us.
"I thought we did really well in the first half, got in their faces and played in their half.
"But we have to be right at it for 45-plus minutes and unfortunately we switched off for the goal."
The Clarets had been the better side in the first half as they chased a first win at Liverpool since 1974 as Barnes and Andre Gray gave the hosts' back four a major test.
But Burnley couldn't hit back after Can's long-range strike and despite matching another of the Premier League's big guns for most of the afternoon, are still waiting for a first away win of the campaign ahead of next Saturday's trip to bottom club Sunderland.
"We had a few chances in the second half. I had a couple and Lowts at the end and if they go in it's a point on the road and everyone's happy," added Barnes, who justified his immediate re-call following a one-match ban with another impressive performance.
"It goes back to the noise that we hear that we haven't won away.
"But that's just noise to us. As a group of players we are confident that we will turn that luck around.
"We need to be sure we have a good week's training and be ready on the front foot and ready for Sunderland next weekend.
"That will probably be harder than today. It will be a completely different game. We need to be mentally switched on and ready to go."