But while the German described the 2-1 Premier League win as "one of the best nights we ever had", he warned his players much better will be required in Saturday's Merseyside derby.
Klopp's men fell behind in the first half as Newcastle's record signing Alexander Isak marked his debut with a goal before Roberto Firmino hauled Liverpool level on the hour mark.
It looked as though they would need to settle for another point in a stuttering start to the season, but eight minutes into added time Carvalho rifled home following a goal-mouth scramble and the roof came off Anfield.
"It's a massive boost from a frustrating night," Klopp said. "One of the best nights we ever had, that's how football is.
"But we have to make sure we use these two days really well. The last two games we got six points, but we have to improve.
"It will not be that Everton open up for us now. We know it will be tough so we have to be ready."
Liverpool had struggled going forward in the first half, with too many passes easily cut out by Newcastle who were then able to threaten on the counter - going ahead in the 38th minute through Isak's impressive strike.
"I said at half-time we were forcing it too early, playing too many long balls," Klopp added. "We looked desperate before we were. Being 1-0 down is obviously not cool but it was the least of my problems at half-time."
The answers to those problems came from two of Liverpool's emerging talents - the 19-year-old Harvey Elliott and Carvalho, who only turned 20 on Tuesday.
"He's now scored twice this week," Klopp added. "Yesterday was his birthday, he's an outstanding player and a wonderful talent, I'm really happy for him.
"Harvey was the best player on the pitch and Fabio decided the game so I'm really happy for the boys."