In a match low on chances, the goals proved decisive in securing new Wanderers manager Owen Coyle's first win since moving from Burnley earlier this month.
However, another good display from South Korean Lee Chung-yong apart, there was little else to cheer.
The standard of play, particularly in the first half, was such that had the thick fog enveloping the Reebok Stadium descended to pitch level the majority of the crowd would probably have been grateful.
United's plans were disrupted before kick-off when centre-back Marcel Seip was injured in the warm-up and had to be replaced by Jon Fortune.
If ever there was a match to pull out before the start then this was it and an indication of the frustration to come was given in the seventh minute.
Lee's teasing cross from the right looked tailor-made for someone like Kevin Davies or Ivan Klasnic but instead the figure of defender Jlloyd Samuel, playing in an unfamiliar left-midfield role, appeared in the six-yard box.
He just failed to reach it with his head but that just about summed up Bolton in the first half.
In the 18th minute Klasnic, with five goals in his last nine matches, brought the first save out of Bunn from Tamir Cohen's cut-back.
However, Blades striker Henri Camara should have done better when he was picked out by James Harper's perfectly-weighted through-ball but he lifted a shot over Jussi Jaaskelainen and the crossbar.
It was a similar story a minute later when Samuel, picked out in the penalty area by Klasnic's clever pass on the turn, cut inside but then tried to get the ball onto his preferred left foot and lost the chance to get in a decent shot.
Even when they were presented with set-pieces in dangerous areas Bolton could not test their opponents as, after the persevering Lee was fouled by Andy Taylor, Klasnic curled in a poor 20-yard effort which was easily cleared.
Klasnic then directed a low 12-yard header so wide it almost went out for a throw-in before being collected by right-back Kyle Walker.
The mood, if not the mist, lifted in the 48th minute when Bolton took the lead with a neat passing move.
Lee's diagonal run and pass to Davies allowed the Wanderers captain to turn and slip a short pass between Chris Morgan and Taylor for right-back Steinsson to run on to and shoot through Bunn's legs from a narrow angle.
It was the Iceland international's first goal of the season and his maiden strike in the FA Cup.
The goal led to a period of sustained Bolton pressure without Coyle's side, currently struggling second-bottom of the Barclays Premier League, really pressing home their advantage.
United manager Kevin Blackwell tried to liven things up by bringing on midfielder Jamie Ward and strikers Richard Cresswell and Ched Evans but the clocked ticked down to their exit without them having much impact.
Six minutes from time substitute Elmander latched onto Lee's scooped pass to fire home the second - in almost identical fashion to Steinsson's effort.
Like their hosts the Blades, sixth in the Championship, also have more important battles to win this season but if defeat taught them anything it was that there was little to separate the bottom of the Premier League from the top of English football's second tier.