The tension created by a goalless opening 45 minutes, in which the brilliant Adnan Januzaj curled a free-kick against the bar and Welbeck missed a sitter, was broken straight after the break when Valencia rammed home from close range.
Welbeck's sixth goal in nine games on the hour made the game safe, although an abysmal miss from Chris Smalling and a goal-line clearance from Leon Britton to deny Shinji Kagawa prevented United adding the extra gloss.
With Sir Alex Ferguson, whose presence at games has been such a talking point, absent on a family holiday, United started with purpose, keen to avoid suffering a fourth straight defeat for the first time since the 1961 team suffered that fate.
Nemanja Vidic volleyed over from close range, although the Serbian evidently felt he should have had a penalty for a push in the back by Jordi Amat that prevented him challenging for the initial header.
Januzaj had taken the free-kick and it is a measure of the progress made by the 18-year-old this season he was also entrusted with another, from a far more central position, shortly afterwards, when Welbeck was sent crashing.
He hit the shot sweetly and Gerhard Tremmel was struggling to reach it.
However, the Swansea goalkeeper was saved by his crossbar, which rattled as the ball bounced away to safety.
The shot proved to be the end of that early United thrust as the visitors pieced together their fluent passing game, denying the home side possession, which in turn made them jittery.
Had Wayne Routledge been able to pick out a team-mate with an intelligent cut-back, United would have been in trouble. Instead, Michael Carrick kept a cool head to tidy up inside his own area.
Patrice Evra brought a decent save out of Tremmel at the other end, with the German goalkeeper also diving low to prevent Welbeck and Valencia reaching Kagawa's low cross.
Tremmel was helpless though when Januzaj and Valencia combined to release Kagawa, who flicked an intelligent cross into the box that should have brought the opener.
Welbeck had enough room to do what he wanted. Bravely he went for the far post but was left turning in despair as the ball rolled narrowly wide.
Welbeck knew what a superb chance it was. Sat high above, so did the injured Robin van Persie.
A sense of anxiety started to take hold, so it was just as well United made that fast start to the second period.
Januzaj created the opener, wriggling free by the touchline before dropping a cross on the six-yard line.
Ashley Williams' attempt to clear led only to the Swansea skipper flicking the ball to Kagawa, whose powerful header was palmed away by Tremmel.
Valencia was ready for the rebound, ramming it gleefully into the net to sighs of relief from the home support.
A vital touch from Williams prevented Evra's low cross from presenting Welbeck with a tap-in, before Kagawa blazed over from 20 yards.
United refused to let their heads drop. And when Evra nudged the ball into the area, after initially looking like he was set to shoot, there was enough momentum behind it for Welbeck to find the corner with a deft flick.
With Januzaj outstanding and Kagawa providing valuable support, some of the hosts' old swagger began to return.
They looked certain to get a third goal but Smalling somehow managed to scoop over from barely three yards after being picked out, completely unmarked, at the far post by Kagawa.
It was the Japan star's turn to fail with the goal at his mercy shortly afterwards when Rafael fed him after leading a rapid break from deep inside his own half.
Kagawa seemed to have done everything right, only for Leon Britton to get back on the line to clear a shot that just lacked enough pace.