A biting wind and persistent rain affected both teams but they still managed to produce some decent football as the playing surface held up well
The Londoners had former England squad goalkeeper David Stockdale to thank for keeping the scoreline respectable as he pulled off a string of saves.
The first chance for Boro came on 14 minutes and should have led to the opening goal, but Cameron Park couldn't apply the touch needed to guide Adam Reach's cross over an unguarded goal-line from close range.
Luke Williams drilled a shot straight at Stockdale as Boro continued to press and Matthew Waters was next to threaten but his header was well saved.
Williams hits the target again with a low drive but the busy keeper was equal to it.
At the other end Cauley Woodrow drew the first save from Jayson Leutwiler as his effort was turned aside and he was denied again when Kyle Oliver blocked on the line from the resulting corner.
The breakthrough came on 26 minutes as Williams tried to find an opening just inside the box and close to the by-line.
As he attempted to escape the attentions of a Fulham defender he was clumsily tripped and the referee pointed straight to the spot.
Andy Halliday, who was again playing at left-back, converted the spot kick in style.
Williams almost made it 2-0 on 32 minutes but Stockdale dived spectacularly to save and hold his shot, before Brobbel fired just wide a few minutes later.
Leutwiler ensuer Boro went into the break level by saving well on 44 minutes to deny Josh Pritchard.
Boro started the second half brightly with Brobbel and Williams shooting wide, while Pritchard scraped the outside of the post for Fulham with an excellent long-range effort.
Cameron Park's free-kick from the left-hand side was saved by Stockdale after Halliday did well to earn a free-kick as he burst through from the wing.
However, a controlled, cushioned header from Williams made sure of the points for Boro as he nodded in a measured Adam Reach cross on 62 minutes.
Boro's next game is at Liverpool on Friday afternoon.
Boro: Leutweiler, Bennett, Halliday, Dolan (Capt), Burgess, Oliver, Park (Fowler 73), Brobbel (Weldon 88), Waters, Williams, Reach (Sirrell 83). Unused subs: Ripley, Edwards.
Fulham: Stockdale, Brister, Musa, Pritchard, Grimmer, Minkwitz, Woodrow, Altman, Banya (Della Verde 76), Islamovic (Tankovic 66), Arthurworrey. Unused sub: Bettinelli.
Referee: Andrew Miller
Man of the Match: Luke Williams. Won the penalty for the first goal and scored the second.