Kris Boyd grabbed the only goal of the game in the 52nd minute to earn a first league victory of the season for the Teessiders, managed by Speed's former Leeds team-mate Gordon Strachan.
Boyd's strike - his first for the club - was one of few memorable moments in a truly dismal encounter, which saw both sides display a disappointing lack of quality in the final third.
United, who parted company with Kevin Blackwell following last weekend's 3-0 defeat to QPR, slip to the foot of the Championship table on goals scored.
Yet they will consider themselves unfortunate not to have claimed a point against a Boro side desperately lacking in creativity and invention.
Former Blades defender Matt Kilgallon and teenage winger Andy Halliday were handed their Middlesbrough debuts, while Estonia forward Tarmo Kink made his first start for the club following two substitute appearances.
Speed made just one change to the United side beaten 3-0 by QPR last weekend, with Nyron Nosworthy starting in place of Stephen Quinn.
Scott McDonald wasted a half-chance for Boro in the eighth minute when he scuffed a tame shot straight at United goalkeeper Steve Simonsen.
The only clear-cut opportunity of the opening period arrived nine minutes later, after Boro's Nicky Bailey had been caught in possession by Jamie Ward inside his own half.
Ward slipped a neat pass through to Ched Evans, but the Welshman's low shot was kept out by the left boot of home goalkeeper Jason Steele.
Ward was involved again in the 24th minute when finding Richard Cresswell in the area, but the former Stoke forward lost his footing as he attempted an over-elaborate turn.
The Boro supporters grew increasingly restless as the half-time whistle approached, although they were temporarily roused in the 33rd minute when Kink fired narrowly over with a dipping left-footed shot from 30 yards.
It proved to be the youngster's final meaningful contribution as he was replaced by Julio Arca at the interval.
The home fans were finally given something to shout about seven minutes into the second half as Boyd opened his account for the club.
McDonald's flick-on was misjudged by Blades skipper Chris Morgan and Boyd broke clear before slotting a low left-footed shot across Simonsen and into the corner.
Speed introduced Daniel Bogdanovic in place of Evans in the 66th minute as the visitors searched for an equaliser.
Boyd was withdrawn from the action soon after, with Leroy Lita coming on in his place.
The changes did little to improve the standard of play as both sides continued to labour in the afternoon sun.
Ward spurned an opportunity to pull United level in the 75th minute when he lashed over from the edge of the area following a neat one-two with Johnny Ertl.
United came close again with five minutes remaining as Ward's shot almost deflected into the path of Bogdanovic six yards from goal.
But Boro, who offered next to nothing as an attacking force after Boyd's goal, held on to claim a somewhat fortuitous victory.