Fletcher handed Hibs the lead in the 10th minute when he latched on to right-wing cross and finished smartly from six yards.
Worringly for Boro, Julio Arca was carried off on a stretcher halfway through the first period following a strong challenge, with the highlight for the Premier League club in the opening 45 minutes being Luke Young's shot which hit the bar.
Stewart Downing fired just over from 25-yards out minutes after the restart, but the Edinburgh side went two ahead when Ross Turnbull misjudged Fletcher's swerving drive.
Gareth Southgate's team then staged a rousing fightback as first Brazilian Alves converted a cross with his first touch after coming off the bench, before David Wheater headed home after a corner was flicked on.
The comeback was complete when substitute Rhys Williams, also with his first contribution, crossed to the far post where Alves nodded home his second.
Boro: Turnbull; Young (Williams 70), Wheater, Huth, Grounds; Aliadiere, O'Neil (Walker 77), Arca (Capt) (Digard 29), Downing (Bennett 70); Tuncay (Johnson 63), Mido (Alves 63). Unused subs: Steele (gk).
Hibernian: McNeil; Van Zanten, Murray, Hogg, Jones (Capt) (Canning 82), Thicot, Keenan (Morais 73), Shiels (Chisolm 73), Fletcher, Nish, O'Brien (Trialist 46). Unused subs: Grof (gk); Stevenson, Campbell.
Still need to sharpen the striker.New keeper for Boro please! Digard looks good in centre.Partner with Shawky or O'Neil.Great to see Alves get his Golden Touch.