Boro enjoyed plenty of early possession and went ahead on 13 minutes when Downing cut inside from the right and fired home with the aid of a deflection off Andy Butler.
Alex Pattison settled the contest with a similar goal in the closing stages when he found space and drilled a shot past Marko Marosi.
Defensively Rovers kept a good shape against their illustrious opponents and can be pleased with their work off the ball.
However, going forward the home side struggled to make inroads and were limited to just two or three chances.
Riccardo Calder was deservedly named Rovers' man of the match after coming on at half time and showing a willingness to be direct and take on his man. He went close to a consolation just moments from full time when his powerful shot was saved by Dimi Konstantopoulos.
Rovers: Etheridge (Marosi 46); Lund (Fielding 68), Alcock, Butler, Garratt (Evina 46); Middleton, Coppinger (Longbottom 46), Blair (McCormick 68), Rowe (Calder 46); Pugh (Mandeville 46), Marquis (Williams 46).
Boro first half: Mejias, Nsue, Morris, Gibson, McGinley, Clayton, Forshaw, Carayol, Downing, De Pena, Nugent.
Boro second half: Dimi, Halliday, Friend, De Sart, Pattison, Baptiste, De Roon, Jackson, Reach, Fischer, Rhodes.
Referee: Darren England