With Joel Matip unavailable and Nathaniel Clyne injured, the 18-year-old was handed his first top-flight start on a big occasion at Old Trafford on Sunday.
And Alexander-Arnold's efforts delighted Klopp, whose side drew 1-1 after Zlatan Ibrahimovic's late header cancelled out James Milner's first-half penalty.
"It was an outstanding performance," Klopp told reporters after the match.
"It is good to have him and show him to our Liverpool supporters. The right-back spot is in good hands even when Clyne is not available.
"Immediately against a player like [Anthony] Martial, one of the quickest players in the Premier League, he needed a few minutes to adapt to the game but then he did very well.
"What a boy. I could have played Milner right and Alberto Moreno left but I didn't want to change two positions. We thought he is ready for it.
"We had a very positive performance. We were the better side, played the better football, had a better plan. We changed the system a little bit and obviously had line-up problems.
"The boys did really well. Unfortunately we only scored with a penalty - there were other situations where we could have been a little bit more clinical to score."
United boss Jose Mourinho and Klopp went head-to-head on the touchline in stoppage-time after a clash between Roberto Firmino and Ander Herrera resulted in bookings for both players.
"I think he wanted a minimum yellow, I don't know," said Klopp.
"It should have been a yellow card just for Herrera. At the end it was a yellow card for the guy who wanted to play football. We could not have the same opinion at that moment."