In the second-round tie at Recreation Park, United's German striker Shala headed the visitors in to a 23rd minute lead from Craig Conway's corner before Goodwillie grabbed the second seven minutes from the break.
The tie therefore was effectively over by the interval and the Tannadice men cruised to the final whistle with Shala and Prince Buaben missing sitters and Alloa keeper David Crawford making a fine save from Shala to keep the scoreline respectable.
The Taysiders had trained on Recreation Park's artificial surface 24 hours before the game and despite five changes from the team which drew 0-0 at St Mirren in the Scottish Premier League at the weekend, looked comfortable enough in the early stages.
In the seventh minute, the visitors created the first half-chance of the game when Conway curled an inviting cross in to the six-yard box from the right but the ball escaped the on-rushing Shala.
The Irn-Bru Second Division side won their first corner in the 18th minute when United's stand-in keeper Steve Banks took no chances as he stretched to tip Mark Brown's long punt over the bar.
However, despite some slack marking from the set-piece Brown headed Scott Agnew's in-swinging cross well wide of the target and the danger was easily cleared.
United's opener five minutes later was straightforward, Shala rising inside a packed six-yard box to head Conway's corner from the left powerfully past Wasps' keeper Crawford.
Even so early in the game, it looked a long way back for the Wasps.
As United began to impose themselves, battling striker Goodwillie set up Buaben at the edge of the Alloa box but the midfielder blazed high over the bar.
In the 38th minute Paul Dixon's searching 50-yard pass sent Goodwillie racing clear of the Alloa defence and he confidently drove in to the home side's penalty area before knocking the ball past Crawford.
The Wasps carried on battling and made a concerted effort at the start of the second half to get back in to the game.
But United were stronger and a yard sharper and it seemed the next goal, if there was to be one, would increase the gap between the two sides.
On the hour mark Buaben had a better effort, this time a low drive from 25 yards, but Crawford got down to his right to save with some ease, before the Alloa keeper made a similar block from Conway moments later.
In the 65th minute Crawford made a fantastic save at his left-hand post from Shala's close-range header and moments later the United striker missed a remarkable sitter, heading Conway's cross on to the turf and over the bar from around three yards.
The home side worked hard in the final stages and striker Andy Scott drew a decent save from Banks with a drive from distance.
Alloa played the last 10 minutes or so with 10 men after right-back David McClune was stretchered off injured after Wasps boss Allan Maitland had used all three substitutes.
With home side on their last legs, Buaben a missed a great chance with four minutes remaining when he hammered the ball past the post from 10 yards when it looked easier to simply pass the ball in to the net.