Rodgers's men have now won only once in six games since the English season restarted last month, but made the most of Eddie Nketiah's red card 15 minutes from time at the Emirates.
Pierre-Emerick Aubameyang's strike gave the Gunners a well-deserved first-half lead, but Kasper Schmeichel in the Leicester goal ensured Arsenal failed to add to their advantage.
And after Nketiah saw red, Vardy slotted home from Demarai Gray's cross to open up a four-point lead on fifth-placed Manchester United, who travel to Aston Villa.
"I think it's huge for us," said Rodgers. "The second half we were outstanding. We were much quicker in our play and even 11 v 11 we were the better team. I was really pleased with how we managed 11 v 10. We got a deserved goal and if we had made the final pass we could have gone on and won the game".