Even with only two of their starting 11 from Saturday's controversial clash at West Brom, the Londoners cruised to success with two goals from Tomasz Radzinski and one apiece for Steed Malbranque and Brian McBride.
These eclipsed a deserved reply for Boston by Lee Beevers which briefly threatened to trigger thoughts of a home revival.
Fulham's silky skills and passing interplay were quickly evident and after a neat build-up, McBride saw a 15-yard shot on the turn palmed aside by Nathan Abbey on two minutes.
In the ninth minute the visitors went ahead when Radzinski accepted a through pass from Malbranque, got in behind the Boston defence and slotted home.
Boston, with Paul Gascoigne involved in midfield, battled manfully to try and hit back but their early efforts yielded just two fruitless corners.
Fulham maintained a lively tempo amid the steady rain and added a second in the 28th minute when Malbranque swapped passes with Luis Boa Morte and then McBride before firing in from the edge of the box.
Six minutes later Boston succumbed again when Malbranque found McBride and with a neat spin 15 yards out he beat Abbey at his near post with a rising drive.
Boston had a couple of close calls just before the break, when first Andy Kirk just missed connecting with an Austin McCann cross and then Mark Crossley bravely dived at Jason Lee's feet as the home striker sprinted goalwards.
Boston made a rousing start to the second half and forced five quick corners, although the culmination of these was only a blocked shot by Lee and an off-target header by Paul Ellender.
But in the 56th minute they got their reward when Ellender dispossessed Adam Green and fired over a great cross for Beevers to head powerfully past Crossley from 12 yards.
For a while the Pilgrims had the Fulham rearguard at full stretch to resist further mounting pressure.
However in the 70th minute one moment of passing brilliance saw Malbranque deliver a 50-yard pinpoint ball for Radzinski to drill beyond Abbey for Fulham's fourth.
Boston went close again in the 85th minute when McCann fired over a cross from the left and a powerful header by Lee was pushed away by Crossley.
Fulham had their evening spoiled two minutes into stoppage time when Liam Rosenior was sent off for picking up his second yellow card of the evening, but it was too late to affect their victory charge.