The impressive Lingard rolled in the opener after 11 minutes, before Welbeck doubled United's advantage before the break in Sydney.
Lingard blasted in a superb third after the All-Stars had reduced the deficit early in the second half, with Welbeck nodding in his second and United's fourth before he was immediately substituted.
Robin van Persie also wasted no time in picking up where he left off last season with the final goal.
A late arrival on United's pre-season tour, van Persie was only introduced for the final half hour after a heavy week's training.
But it was long enough to claim the assist for Welbeck's second before atoning for an uncharacteristic miss by driving home from close range almost immediately afterwards.
After the false start in Bangkok last weekend, it completed a satisfactory evening for the Red Devils, who fly to Japan on the next leg of their tour on Sunday for matches in Yokohama and Osaka.
And for Moyes it was a welcome first win, with Warrington-born youngster Lingard having the honour of scoring United's first goal for their new manager in front of 83,127 at ANZ Stadium.
That an Australia national side missing its European contingent was facing South Korea in an East Asian Cup encounter at virtually exactly the same time as the remainder of the A League was in action underlined the scratch nature of Ange Postecoglou's team.
Little wonder they struggled defensively, with Welbeck exchanging passes with Lingard before firing an early chance straight at Ante Covic.
It was not long before the same pair were combining to score the first goal of Moyes' time in charge.
Showing imagination to dummy Ryan Giggs' pass, Welbeck allowed the ball to run on to an unmarked Lingard.
The 20-year-old kept his nerve to put the visitors in front from 10 yards.
Thomas Broich fizzed a shot over from the edge of the area but this never looked like being a repeat of last Saturday's stuttering display.
A delicate pass from Giggs sent Welbeck beyond the All Stars defence just past the half hour and the England forward scored.
After netting only twice last season, Welbeck needs to make a positive start this term, especially as Wayne Rooney's future is open to such debate and Javier Hernandez is going to miss the remaining three games in the Far East.
Welbeck was a man on a mission and Michael Beauchamp denied him with a brave sliding block before the forward blasted into Covic but the ball ricocheted wide off the keeper's legs.
The hosts improved after the break, with Besart Berisha pulling one back with an understrength shot Anders Lindegaard was unable to prevent creeping in at the corner.
United's reply was instant, and yet again Lingard supplied the finish, a brutal effort from 12 yards.
Welbeck turned home Van Persie's corner not long after the Dutchman's introduction and he almost completed his hat-trick with a shot that rolled narrowly wide.
But it was left to Van Persie to have the final say, the back-to-back Golden Boot winner taking no time to get into the groove once more.