Rodgers suffered a last-gasp defeat in his maiden game at the helm last weekend at Watford and his side had to work hard to secure maximum points against a spirited Fulham side, which is now 13 points from safety.
On-loan Monaco midfielder Youri Tielemans was the Foxes' stand-out player and he got them on their way with a simple finish midway through the first half, only for Floyd Ayite to pull the Cottagers level with their first effort on target in the 51st minute.
As the Foxes looked to be heading for a frustrating draw, Vardy powered through and unleashed an unstoppable drive into Sergio Rico's bottom-right corner to reach his personal landmark - then followed that up with another cool finish four minutes from time to wrap up all three points.
Other Premier League results:
Huddersfield 0-2 Bournemouth
Southampton 2-1 Tottenham
Cardiff 2-0 West Ham
Newcastle 3-2 Everton
Manchester City 3-1 Watford