"The first target was to get through this group. I didn't expect we would be through after four games but the boys did it and it's really well deserved," said Liverpool boss Klopp.
Atleti were handed another setback before the break when Felipe was shown a red card for a foul on Mane, who appeared to have a straight path to goal, leaving the LaLiga club to finish the match with 10 players.
Diego Simeone's Atleti tried to get back in the game, but were left frustrated as Liverpool dominated possession and saw out the result to take all three points.
First-placed Liverpool are on 12 points in the group and are a lock to reach the round of 16 having won four games in four tries, while Atleti are in third with four points behind Porto, who sit second after earning a draw with bottom side AC Milan earlier in the day.
A draw would have been enough for Liverpool to be sure of their passage to the last 16 but from the outset Klopp's side looked like they had nothing but victory on their minds.
Jota put the home side ahead in the 13th minute, nodding home at the back post after a great cross from the right by Alexander-Arnold.
Alexander-Arnold was again the provider for the second eight minutes later when his low ball into the box was brilliantly met by Sadio Mane who fired past Jan Oblak.
The Spanish side's troubles intensified after Felipe was sent off in the 36th minute for a deliberate trip on Mane that earned him a harsh straight red.
Jota thought he had his second and Liverpool's third, shortly after the interval, but the goal was ruled out for offside after a VAR review.
Former Liverpool striker Luis Suarez also had a deflected effort, which beat Alisson Becker, ruled out by VAR for offside but the Spanish champions could have no complaints about the outcome.
The only negative for Klopp's side was a hamstring injury to Brazilian forward Roberto Firmino in the second half.