On-loan keeper Ryan Whitley failed a late fitness test, but Lewis Maloney returned to the side after missing the previous three games.
There was further injury woe for Boro when Michael Coulson limped off after just 10 minutes, and the visitors took the lead from their first attack five minutes later when a ball across the Boro goalmouth was slammed into the top corner by Rob Apter.
Boro came close to equalising when a Maloney free-kick was tipped over by visiting keeper Mateusz Hewelt, then a Ryan Watson shot was easily held by the keeper.
A terrific move involving Nick Hutton, Nathan Cartman and Watson ended when the latter’s shot was saved, then a long kick upfield by veteran keeper Michael Ingham bounced up over Hewelt and against the bar, but Bamber went 2-0 up on 36 minutes as Isaac Sinclair latched onto a through ball and curled his shot into the bottom left corner of the net.
Boro reduced the arrears on the stroke of half-time, when Will Thornton battled for possession on the left, and sent a pass into the box for Lewis Turner to strike a shot that took a deflection and looped up over the keeper.
Early in the second half a low cross from Ash Jackson was flicked inches past the post by the hard-working Cartman, but the visitors looked dangerous every time they went forward, and they increased their lead in the 68th minute following a rampaging 30-yard run from 18-year old winger Rob Apter, on loan from Blackpool, who passed to Isaac Smith, and his cross was headed home by 19-year-old striker Ewan Bange, also on loan from Championship club Blackpool.
Maloney then sent a free-kick past the far post as Boro tried to fight back.
Bamber went 4-1 up on 76 minutes with a terrific individual goal from the outstanding Apter, as he latched into a pass, raced forward and blasted a thunderous shot past the helpless Michael Ingham.
Boro looked for some late consolation, but the well organised Bamber defence stood resolute until the final minute, when Turner fed Jackson, who cut in from the left wing and fired in a strong shot that was pushed round the post by Hewelt, but the resulting corner fell to James Cadman on the edge of the area, and he swivelled to lash home a stunning right-foot volley into the bottom corner to make the final scoreline 4-2.
Another disappointing result, but Cartman ran his heart out up front, Will Thornton was excellent at the back, and there were decent showings from the likes of Cadman, Kieran Weledji and Ash Jackson.
BORO: Ingham, Gooda, Jackson, Maloney, Thornton, Cogill (Cadman 65), Hutton (Hudson 61), Watson, Coulson (Weledji 10), Turner, Cartman.
BORO MAN OF MATCH: Nathan Cartman.