Four of the goals were scored in the opening 45 minutes at Ibrox Stadium, before Odsonne Edouard netted what proved to be the winning strike 21 minutes from time.
Celtic, who twice trailed and played more than a third of the match with 10 men due to Jozo Simunovic's red card, have now won on enemy territory four times in a row for the first time since 1909.
After seeing his side move nine points clear of Rangers at the top of the Scottish Premiership, Rodgers told reporters: "You always have to have humility in your approach to the game and respect your opponent.
"There was a lot of noise before the game, but that's natural - they've been on a good run and at home, thinking they could get the victory.
"But I think the players today showed their quality and that real champion mentality. It's a big step for us to go nine points clear with a game in hand, but we still have a lot of work to do. But an absolutely brilliant win and a great day for the supporters."
Rodgers has not been on the losing side against Rangers since being appointed by Celtic in the summer of 2016.