Brazilian Marcos Rhoden made another start while his compatriot Rafael Peieira was named on the bench for the first time, after recovering from injury.
The home side opened the scoring after 12 minutes when a long ball over the top put Cauley Woodrow in the clear. He strode into the area and side-footed it calmly past Connor Ripley.
The lead should have been doubled minutes later when Omri Altman saw his shot deflected wide. From the resulting corner, Christensen saw his point-blank header bounce into ground and then over the crossbar.
On 22 minutes Boro created their first real chance when Cameron Park swung over a free-kick from the right-hand side. It was met first time by young striker Bradley Fewster, but his shot went well over the bar.
Half-time: Fulham U21s 1 Boro U21s 0
Boro started the second half much better and nearly equalised three minutes after the restart. Fewster broke down the right and saw his cross contested by Luke Williams. The ball fell kindly to Ryan Brobbel who set himself but his shot went inches wide of the far post.
The last 20 minutes saw Boro pushing strongly for the leveller. On 70 minutes Park saw his goalbound shot knocked behind. From the corner Adam Jackson, who was impressive all night, saw his header cleared at the back post.
Four minutes later another Park free kick was fizzed in. Again Jackson got his head to it, only for the keeper to pull off a smart stop.
Fulham nearly put the game out of reach 13 minutes from time but Dino Islamovic saw his diving header go wide of the target.
Boro were piling forward now and came oh-so-close to scoring in the 84th minute. Captain Christian Burgess saw his towering header from yet another Park set piece go millimetres wide.
Boro: Ripley, Bennett, Sirrell, Morris, Burgess (c), Jackson, Brobbel, Rhoden (Kneeshaw 53), Fewster, Williams (Peireira 74), Park. Unused subs: Fryer, Okosieme, Waters
Fulham: Bettinelli, Passley, Kavanagh, Christensen, Grimmer (c), Arthurworrey, Altman, Minkwitz (Banya 71), Woodrow (Islamovic 46), Williams (David 59), Tue Na Bangna. Unused sub: Joronen
Booked: Morris (Boro), Christensen (Fulham) - both fouls
Boro Man of Match: Fresh from a loan spell at Halifax, the returning Adam Jackson was immense all night and could easily have scored a couple of goals too.