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Smith had already headed a John Oster cross straight at goalkeeper Paddy Kenny when he fired his side in front by rounding off a flowing move.
George McCartney accepted Mart Poom's throw out and exchanged passes with Julio Arca before picking out the striker on the edge of the penalty area.
He took one controlling touch before thumping a left-foot shot across Kenny and into the back of the net, and the pattern for the first half was set.
Chances, however, were few and far between as the Black Cats continued to dominate, although Jason McAteer sent a looping header just wide four minutes before the break.
For their part, the Blades created only two openings of note, both falling to Michael Tonge, who drilled an eighth-minute free-kick narrowly off target and then fired wastefully over 13 minutes before the break.
Neil Warnock knew he had to change it at half-time and sent on Paul Peschisolido and Alan Wright for Andy Parkinson and Stuart McCall, and his troops responded quickly.
Chris Morgan got his head to a Tonge corner and Phil Jagielka just failed to connect in front of goal after 48 minutes.
Wayne Allison saw a shot blocked by Gary Breen and Peter Ndlovu missed his kick in front of goal after Allison had headed down a Stephen Wright cross.
The travelling fans sensed a resurgence, but it might have been killed in its infancy in the 56th minute when Smith got away from Robert Page and fed Oster before heading into the box to collect his cross on his chest, only for Morgan to intervene.
Sunderland, having been rock solid in the first half, looked decidedly unconvincing at the back, but as the storm abated, they regained their composure.
Ndlovu skied a shot on the turn after 66 minutes, but as United poured forward, they left gaps at the back which the home side, with a little more precision, could have exploited.
McCarthy's men were dealt a blow with 15 minutes remaining when Arca was stretchered off, and they were appealing in vain for a penalty three minutes later after Kevin Kyle went down under Morgan's challenge, but referee Steve Dunn waved play on.
However, they saw out the remaining minutes - including more than six for stoppages - with some comfort .