Goals from Roberto Firmino and Sadio Mane saw Liverpool take control at Anfield in the opening 45 minutes before Danny Welbeck pulled one back after the break.
Georginio Wijnaldum ensured Liverpool - soundly beaten 3-1 at Leicester City on Monday - emerged victorious with a goal well into stoppage time to end Arsenal's hopes of salvaging a draw.
"It was one of the best games we have played so far because the strength and the line-up of opponent," boss Klopp told BBC Sport.
"We did really well. All of them played a fantastic game. When we are compact it is fantastic. 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.
"We really felt bad last week, we needed a few days to understand what happened. We had hard words after the defeat at Leicester. We analysed it and that was not enjoyable. We had another opportunity and we took it. It is the roller coaster of the Premier League. It is not always going up.
"We all know what we did last week and how we played. It is not usual to come back like this. We know what we need to work on.
"It is important to go back to fourth above Arsenal."