The 24-year-old striker netted twice in Watford's opening day win at Norwich and repeated the feat against the League Two side, scoring either side of Marvin Sordell's effort at The Rec on Tuesday night.
Manager Malky Mackay made just one change from Friday's Championship opener.
Rene Gilmartin replaced Scott Loach in goal after the 22-year-old received a late call up to Fabio Capello's England squad for Wednesday's friendly with Hungary.
After some early Aldershot pressure, Watford took control, with Graham's first coming on 15 minutes when he was played through on the edge of the box by Sordell and calmly slotted the ball low beyond goalkeeper Jamie Young.
The second came on the stroke of half-time as Sordell switched from provider to poacher to get on the end of Lloyd Doyley's cross to poke home with the outside of his foot.
Graham got his second three minutes after the break as he beat the offside trap fire the ball into the top corner from 20 yards.
Aldershot went close to pulling one back through Damien Spencer in the 88th minute, but Mackay's men easily progressed.