The Magpies put on another solid display and took the lead in the 38th minute through debutant Alexander Isak. The Reds improved in the second half and found the equaliser through Roberto Firmino.
Newcastle, who held Manchester City to a 3-3 draw just 10 days prior, appeared to be heading for another point when birthday boy Fabio Carvalho fired in the winner late in stoppage time.
Speaking after the match, manager Eddie Howe told BT Sport: "I thought we got a really good performance.
"In the first half we were very brave, pressed them high, tried to disrupt their rhythm, in a different way to our game against Manchester City.
"I thought we defended well for large parts, minimised their chance to play through us. I was very pleased with the commitment and unity, and bitterly disappointed with the end."
Five minutes of added time had been indicated, but Howe said he would leave it to others to debate why eight were actually played.
"As I am sat here I have no idea," he said. "It seemed the final whistle would never come but we needed to defend the final situation better than we did. It is a cruel game."
Amid the frustration there was encouragement in the performance of Isak, who might have had a second goal but for a tight offside call before leaving the game with a dead leg.
"His finish and the overall performance was very good," Howe said. "I'm really pleased with his tactical discipline.
"On the ball he gave us that pace in behind that teams need. The second goal I didn't realise how close it is. He showed what he will bring the team."