Boro, on the other hand, have now lost six of their last seven matches, and Strachan has only two victories to his name in the 12 games he has taken charge of since being appointed in October.
New signings Barry Robson, Willo Flood and Chris Killen were all included in Boro's starting line-up after leaving Celtic this week to join up with Strachan for a second time.
Matt Kilgallon was a notable absentee from the Blades' starting line-up, with the sought-after central defender seemingly set for a move away from Bramall Lane.
Neither side looked like candidates for promotion in a dire first half and it took until the half-hour mark for Boro midfielder Isaiah Osbourne to register the first shot on target.
The on-loan midfielder burst into the penalty area, but his scuffed strike was never going to trouble Mark Bunn.
Scotland international Robson was showing some nice touches and he swung in a free-kick from the right which Chris Riggott rose highest to meet but Bunn again made a regulation save eight minutes before the break.
United were booed off at half-time after a poor first-half showing, in which they failed to test Boro keeper Danny Coyne.
The Blades were not better at the start of the second half, but they took the lead from out of the blue nine minutes after the break.
Andy Taylor was the architect, swinging in an inviting cross from the left which Cresswell headed home from six yards out after shrugging off David Wheater.
Boro were unfortunate to be behind and they felt aggrieved not to be awarded a penalty when Jeremie Aliadiere went down under a challenge from Chris Morgan, but referee Andy D'Urso waved away the protests.
Strachan introduced Adam Johnson with just over half an hour remaining and the highly-rated showed the touch of class Boro had been lacking when he picked out Justin Hoyte with a clever pass and the full-back crossed for Robson, who fired over under pressure from two defenders.
United should have doubled their lead after 68 minutes but James Harper volleyed over from eight yards out.
And Bunn produced a fine fingertip save to divert Henri Camara's strike onto the post six minutes from time.
Boro strived for an equaliser late on, but United held on for three precious points.