The Dutchman endured a frustrating spell last season after a move to Swansea foundered but shone alongside Lukas Jutkiewicz up top for Tony Mowbray's side.
Former Stoke midfielder Dean Whitehead also impressed on his home debut, starting the move that Emnes finished from close range in the 20th minute to put Boro ahead.
Emnes pounced again five minutes after the break, tucking home the rebound when Kevin Olimpa - a half-time change for Cedrid Carrasso in the Bordeaux goal - made a mess of his first task, spilling a Jutkiewicz shot.
The French side belatedly upped their tempo, Hadi Sacko twice hitting the goalframe before Henri Saivet found the top corner of Jason Steele's net with a curler (85).
But Boro deserved the morale boost ahead of next weekend's Championship opener against Leicester and Faris Haroun might have provided a final flourish with a volley just off-target.