Tony Mowbray's side looked bright in the first-half and went ahead just before the break when Kalifa Cisse turned Leroy Lita's cross into his own net.
Lita doubled the visitors' lead against the club where he began his senior career before Barry Robson made it three with a superb solo effort 13 minutes from time.
It got worse for the Robins in the final minute when goalkeeper David James was shown a straight red card for hauling down substitute Cameron Park in the area.
With all their substitutes used, defender Liam Fontaine went in goal for the penalty which Lita slotted away with aplomb.
The victory extended Boro's unbeaten league run to four matches and moved them four points clear of the bottom three, only two points behind City.
Middlesbrough made an enterprising start and Gary O'Neil left fly with an effort from the edge of the area but James made a good save.
The home side responded and Brett Pitman looked to be clean through on goal after a mistake from Joe Bennett, but opted to take the shot early and it went over.
The game descended into a war of attrition as the half went on with neither side able to fashion a meaningful goalscoring chance.
Boro suffered a blow with half-time approaching when Justin Hoyte was forced off injured but the visitors took the lead two minutes before the break.
Lita got away down the left and drilled in a low cross which Robins midfielder Cisse could only direct into his own net.
Despite suffering that blow at the end of the first half, City started the second half well with Marvin Elliott bringing a decent save out of Steele.
However just as it looked as if they might get back in the game, the home side found themselves 2-0 down with 58 minutes gone.
Tony McMahon whipped in a deep cross which the City defence failed to deal with and Lita headed in at the far post.
Middlesbrough could have been further in front three minutes later but Robson headed wide from close range when perhaps he should have done better.
With 13 minutes remaining Robson made amends and put the game to bed when he skipped past a couple of challenges before sending a precise chip past James.
That sparked a mass exodus of City fans from Ashton Gate and it got worse for those left in the ground in the final minute.
James saw red for bringing down Park and Lita fired past stand-in goalkeeper Fontaine to inflict the Robins' heaviest defeat for the season.
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