Goodwillie all but settled the tie with United's second goal
First-half goals from Andis Shala and David Goodwillie gave Dundee United a comfortable and expected win over Alloa in the Co-operative Insurance Cup.
Striker Shala headed the Scottish Premier League side into a 23rd-minute lead from Craig Conway's corner.
Goodwillie grabbed the second seven minutes from the break after racing clear following Paul Dixon's fine pass.
Shala, who was also denied by a good David Crawford save, and Prince Buaben missed chances to add to the score.
United had trained on Recreation Park's artificial surface 24 hours before the game and, despite five changes from the team that drew 0-0 at St Mirren in the Scottish Premier League at the weekend, looked comfortable enough in the early stages.
Conway curled an inviting cross in to the six-yard box from the right but the ball escaped the on-rushing Shala.
Some slack marking allowed the Second Division side a chance after 18 minutes, but Mark 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.
United were in control from the minute Andis Shala opened the scoring
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.
However, the tie was all but settled when 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.
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 Steve 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 tiring, Buaben 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.
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