The 21-year-old, who only joined League Two Boro on loan from Watford on Friday, curled a free-kick in off the inside of Seb Brown's post after 24 minutes to set the visitors on their way.
Luke Moore wasted a glorious chance to level before sub Yemi Odubade booked Stevenage's place in the third-round draw, while Dons' Ed Harris was dismissed for two yellow cards.
Boro went ahead after 24 minutes when Walker fired a 20-yard free-kick in off Brown's post with the goalkeeper caught flat-footed.
Brown reacted well to keep Conference outfit Wimbledon in the contest when he clawed Lawrie Wilson's strike over the bar after 55 minutes.
Luke Moore should have levelled when he headed Danny Kedwell's cross over from six yards before Odubade struck in the 81st minute and Harris collected his second yellow for a high foot in the build-up to the goal.
Source: PA
Source: PA