The Portuguese attacker celebrated his 20th birthday on Tuesday and came off the bench to complete Liverpool's comeback from a goal down after they looked to be heading for a point.
"It feels like this is the best way to win a football game -- to be honest, I don't need it every week, but if you can do it once or twice it's really good," Klopp said following the victory. "I told the boys at half-time even before we went 1-0 down we looked a bit desperate in situations when we didn't play the extra pass. When we did that we were immediately really dangerous.
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"We had to force ourselves to keep calm a little bit more and to keep going, and in the end we forced it."
The hosts, fresh off a 9-0 thrashing of Bournemouth that led to the sacking of Cherries coach Scott Parker, went into half-time trailing 1-0 after the Magpies' new arrival Alexander Isak smashed a close-range shot into the roof of Alisson's net just before the break.
Liverpool levelled the score on the counter right at the hour mark when Roberto Firmino beat Nick Pope with a first-time shot after being picked out by Mohamed Salah in the centre of the penalty area.
Isak's debut came to an end three minutes after the equaliser when he was replaced by Chris Wood, and the visitors struggled to get the ball up to him as Liverpool pressed forward relentlessly, peppering their goal with shots.
Newcastle defended stoically, but Carvalho made the most of their failure to clear a corner, thumping his shot into the roof of the net to snatch a win that lifts Liverpool to fifth in the table on eight points, with Newcastle 11th on six points.
"We've shown up today for the fans, they've helped us so much, I want to say a big thank you to them because without them this wouldn't be possible," Carvalho said after the match.
After seeing his side concede one of the latest winners in Premier League history, Newcastle manager Eddie Howe described the result as "painful."
"I don't really remember them having too many clear-cut chances," Howe said. "Yes, they had pressure, they're long shots, but I thought we defended really well. The last goal is really a heartbreaker for us."
Information from Reuters was used in this report.