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, who are 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.
After dominating the early stages, Leicester forged ahead in the 21st minute. Wilfred Ndidi released Jamie Vardy after picking Calum Chambers' pocket in the middle of the park, the striker teeing up Tielemans, who took a touch before stroking into an empty net.
Harvey Barnes then fired over, while James Maddison flashed wide from distance as the hosts threatened to run riot.
Fulham caretaker boss Scott Parker's decision to replace Ryan Sessegnon with Ayite at the interval paid off within six minutes. After beating Ben Chilwell to Havard Nordtveit's long pass on the right touchline, the Togo international surged into the penalty area and beat Kasper Schmeichel with a deflected shot.
A superb Rico save denied Barnes' whipped effort shortly after the hour mark before a full-stretch Schmeichel tipped Aleksandar Mitrovic's header past the post.
The stage was then set for Vardy. The striker reached 100 goals in all competitions for the club with a typically direct run and unerring finish 12 minutes from time, while he added another to his impressive tally by scuffing home Barnes' pull back in the 86th minute.
What does it mean? Rodgers up and running, Fulham doomed
Rodgers spoke about his excitement at working with Leicester's talented young squad upon his arrival from Celtic and they certainly put on a show for him in the first 45 minutes.
They dropped off at the start of the second half, allowing Fulham back into the game, but their elder statesman did what he does best in the final 15 minutes to secure successive home wins for the Foxes.
Fulham, meanwhile, need a miracle – and a significant improvement in their defending – to avoid a quick return to the Championship.
Tielemans lights up Foxes' midfield
The Belgian has impressed since arriving in a swap loan deal with Adrien Silva in January and added to his growing reputation with another fine performance. Not only did he open his account for the club, he routinely bamboozled Fulham with inventive passing and incisive movement.
Chilwell endures a game to forget
The young England international will hope rumoured suitors Manchester City do not look too closely at this performance. His positioning was wayward throughout and he should have done more to deny Ayite for his equaliser at the start of the second period.
Key Opta facts
- Vardy scored his 100th goal for Leicester City in all competitions, becoming the first player to achieve that feat for the Foxes since Gary Lineker.
- Fulham have lost 14 of their 16 Premier League away matches this season (W0 D2 L14), conceding 40 goals.
- Vardy has scored 74 Premier League goals - since the start of the 2014-15 season, only Harry Kane (122), Sergio Aguero (109) and Romelu Lukaku (81) have scored more.
- Leicester boss Rodgers has won all seven of his Premier League matches against Fulham. It is the joint-best 100 per cent record in the competition's history for a manager against a single opponent (also Jose Mourinho v Portsmouth and Roberto Mancini v Wigan Athletic).
- Fulham's Ayite became only the second Togolese player to score a Premier League goal, after Emmanuel Adebayor (97 goals between 2006 and 2016).