Liverpool win back-to-back games in the Premier League for the first time this season.
The visitors had earlier taken a deserved lead as debutant Alexander Isak finished powerfully in an evenly contested first half on Merseyside.
Isak had the ball in the net again early in the second half after good work on the counter attack but was marginally offside, and Liverpool were level shortly after as Firmino ran onto a Mohamed Salah pass from the right to sweep a shot into the bottom-left corner of the net.
Eddie Howe’s men still looked good value for at least a point even after Roberto Firmino sidefooted home just past the hour mark, with that blow being followed by some impressive defending.
Newcastle had at times leaned on the dark art of time-wasting to halt the Reds’ momentum, and that came back to haunt them at the death.
Having initially signaled five minutes of added time, referee Andre Marriner allowed eight to be played, and it was the last of those that saw Carvalho smash the ball in off the bar after a scramble from a corner.
It was a result that prompted scenes of sheer elation at Anfield and hopes that a mixed start to the season can be consigned to the past.
As for Newcastle, they were left to grapple with the bitter taste of defeat for the first time this term, although they will take heart from another impressive showing.
Jurgen Klopp gives his verdict to BT Sport. “It was a perfect moment. It was a difficult game. Very passionate, aggressive defending opponent, and it is difficult to pick up the rhythm. Even before we were 1-0 down we looked a bit desperate, so we had to force ourselves to be calm. It was a wonderful goal from the birthday kid. This is the best way to win a football game. I don’t need it every week, but once or twice is really good! We scored on 90 plus eight is the perfect response [to Newcastle’s delaying tactic] and I am really happy about that. But nothing else to say. It is massive, it’s great, we will remember this for years and years.”