Goals from Roberto Firmino and Sadio Mane gave Liverpool a comfortable 2-0 half-time lead before Alexis Sanchez - dropped by Arsene Wenger - made his belated introduction to the match immediately after the interval.
He laid on the assist for Danny Welbeck, who was making his first Premier League start of the season, to halve the deficit and inject some much-needed pace into a lethargic Arsenal side before Georginio Wijnaldum sealed all three points in stoppage time.
Despite Liverpool's two-goal cushion at the end of the game, Mignolet had to be alert to make some crucial saves as Arsenal rallied in the second - including a fine stop to tip an Olivier Giroud effort onto the bar.
"All of them played a fantastic game," Klopp told BBC Sport. "When we are compact it's fantastic.
"Adam Lallana can come out of the formation and trigger something. Being compact and stable is the basis of each good display.
"We knew Arsenal would bounce back in the second half. Alexis Sanchez is the highest quality player and plays different to Danny Welbeck. Simon Mignolet saved our lives.
"It's important to go back to fourth above Arsenal. We really felt bad last week, we needed a few days to understand what happened."