Hodgson put pen to paper on a new 12-month rolling deal earlier in the week and saw in-form striker Bobby Zamora help shoot down League One side Swindon Town at Craven Cottage.
Zamora broke the deadlock after just 16 minutes of the tie with his 11th goal of the season and sixth strike in the last seven games.
The returning Andrew Johnson missed a second-half penalty on his first start since picking up a groin injury in October, but the Premier League side deserved to progress after carving out the game's clearest chances.
Hodgson had named a strong starting line-up and Zamora should have opened the scoring with just two minutes on the clock when he was picked out by Bjorn Helge Riise only to fire wide from ten yards after making space with a neat dummy.
However, he made amends 14 minutes later when he latched onto defender Stephen Kelly's long pass and calmly lifted the ball over advancing goalkeeper David Lucas.
Johnson was inches away from converting Paul Konchesky's cross on the half-hour mark.
Fulham did not have it all their own way as Billy Paynter lashed a shot over Mark Schwarzer's crossbar towards the end of the first half before teeing up Jonathan Douglas who dragged his effort wide of the target.
Zamora should have made it 2-0 to the Cottagers on the stroke of half-time but looked on forlornly after heading Jonathan Greening's pinpoint cross over the bar from just six yards.
Roared on by 6,500 travelling fans, Swindon made the better start to the second half with both the lively Paynter and Alan Sheehan testing Schwarzer with snap-shots.
However, the visitors were let off the hook in the 63rd minute when Johnson missed a penalty he had won after being felled in the box by Lucas.
The Robins keeper was lucky to only see yellow for his challenge and rubbed salt into the home fans' wounds with his ensuing save from a tame spot-kick.
Lucas was enjoying his afternoon and denied Clint Dempsey on 70 minutes when he raced off his line to smother an effort from the USA international.
Johnson refused to let his head drop and came close to scoring his first goal since August when he let fly from distance with a shot which whistled just wide of the target.
But Swindon failed to seriously test Schwarzer in the final ten minutes as the home side held on for a straightforward cup victory to maintain their impressive recent run of form.