Goals from Liam Parrington and Marce Collington secured the points for Kevin Keen's team, who came from behind to make it three wins in a row over capital opposition following their victories against Charlton Athletic and Tottenham Hotspur in recent weeks.
After starting poorly, the Hammers conceded a freak goal on 15 minutes when a deep free-kick from the right missed out everyone in the penalty area and dropped in at the far post.
That provided a wake-up call for the hosts, who fought back and forced the Fulham keeper into two magnificent saves from Collington and Tony Stokes, before the equaliser arrived 10 minutes before half-time, when Danny Bunce's corner was bundled over the line by Parrington.
Hammers stepped up a gear in the second half and scored what proved to be the winner on 65 minutes when the impressive Collington rounded off a mazy run with a low right-foot drive into the corner of the net.
The lead could have been extended on two or three occasions in the remaining 25 minutes, while at the other end goalkeeper Matt Read was forced to make two good saves with his legs as Fulham searched in vain for an equaliser.
Said Kevin Keen afterwards: "The players started very poorly and we conceded a terrible goal, which was particularly disappointing after we had defended so well in recent games and kept two clean sheets.
"That gave them a bit of a shock and we started to get our game together after that. In the second half we played a lot better and could really have added to the second goal.
"Overall, it was very pleasing to come from behind and secure the win, and hopefully this winning run will continue."
Team: Reed, Henry, Parrington, Blewitt, Cohen, Wright, Tucker, Noble, Bunce, Collington, Stokes (O Allen 60). Subs unused: Singh, Ketchell, Behcet, Yao.
Meanwhile a Billy Mehmet penalty wasn't enough for Tony Carr's side against a Fulham team who were forced to play with 10 men for the last 30 minutes of the game.
In blustery and difficult conditions in west London, both sides found it hard to find any passing rhythm but it was Fulham who took the lead just minutes after the interval, when a slip from under-19s debutant Tom Laws, who otherwise put in a very impressive performance, gifted them a goal.
Hammers responded on the hour when James Allen's corner looked set to directly hit the net until a defender cleared the ball using his fist on the line. The referee immediately spotted the infringement and dismissed the offender, leaving Mehmet to convert from the penalty spot.
However, try as they did, the visitors couldn't take advantage of their one-man advantage as Fulham dug in and defended resolutely for the remaining half an hour.
Greg Pearson should have scored when he raced clear following a defensive mix-up, but the striker fired his volley over the bar, and Hammers were forced to settle for a point against their nearest rivals in the Academy League group C table.
Said Tony Carr: "It wasn't the best of matches, mainly due to the windy conditions, and I am probably grateful that we secured a point at the end of the day.
"Fulham have a very good side and caused us a few problems at times. Young Tom Laws, who is only a first year scholar and was playing his first game at this level, was sadly punished for his only error in the whole match, but we fought back well and deserved the equaliser.
As often happens, though, going down to 10 men gave them a bit of extra determination to get something out of the game with the odds against them, and they defended like trojans for the last half an hour.
"The important thing is that we haven't lost, because the draw maintains the five-point cushion we have over Fulham, who are behind us in the league table."
Team: Jackson, Khan, Tattam, Ferdinand, Laws, Lumsden, Allen, McMahon, Mehmet (Carrick 78), Pearson (Eastwood 78), Cole (Akinsete 62). Subs unused: Fletcher, Morris.