The £1million summer buy from Norwich struck low into the bottom left corner after 42 minutes following a spell of dominance for the home side.
Fellow new boy Jonathan Hogg had gone 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 defensive hero after Clayton beat two Bradford defenders in the box from Jack Hunt’s pass.
But there was nothing Darby could do to prevent the opener just seconds later when Hunt again supplied the ammunition and Vaughan fired in from long distance.
Vaughan, Huddersfield’s 14-goal top scorer last season while on loan from the Canaries, then 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.
Garry Thompson had fired wide after 15 minutes while home goalkeeper Alex Smithies saved Raffaele De Vita’s header from Kyel Reid’s corner four minutes later. Thompson and then Reid were denied after Hunt gifted the ball to Alan Connell while trying to clear De Vita’s free kick.
Huddersfield had gone close through Martin Paterson twice and Anthony Gerrard before breaking the deadlock.
Visiting manager Phil Parkinson opted to change things after 63 minutes with an attack-minded triple substitution when strikers Naki Wells and James Hanson, rested from Saturday’s 2-2 draw at Bristol City, came off the bench along with James Meredith.
Bradford, backed by 4,000 travelling fans, had not scored against the Terriers in four out of their last five derby meetings, but Wells sparked late drama in the 90th minute with a 20-yard rocket for a goal that gave them hope.
Huddersfield nearly shot themselves in the foot in stoppage time when Smithies flapped at a corner, but Hanson was crowded out before he could pounce.