The £1million summer buy from Norwich struck low into the bottom left corner after 42 minutes after a spell of dominance for the home side.
Fellow new boy Jonathan Hogg went close with a similar low shot two minutes earlier and then Adam Clayton had an effort cleared off the line. Stephen Darby was Bradford's hero after Clayton beat two Bradford defenders in the box from Jack Hunt's pass.
But there was nothing he could do just seconds later when Hunt again supplied the ammunition and Vaughan let rip from distance.
Vaughan, Town's 14-goal top scorer last season on loan from the Canaries, settled the tie after 54 minutes. Winger Adam Hammill, who replaced Hunt at the break, crossed and the unmarked Vaughan buried a powerful header past goalkeeper Jon McLaughlin.
Bradford, who beat Wigan, Arsenal and Aston Villa on their way to Wembley last term, had chances throughout.
Thompson cut through the Town defence on 57 minutes but could only fire straight at Smithies when clear on goal.
Substitute strikers Nahki Wells and James Hanson, rested from Saturday's 2-2 draw at Bristol City, came off the bench and Hanson went close from Reid's cross after 78 minutes.
Wells sparked late drama in the 90th minute with a 20-yard rocket that had Bradford back in it but the Bantams could not find another goal.
- Adam Clayton
- Jack Hunt