Booth followed up a Musselwhite spill from a deflected Chris Hay shot in the 38th minute to grant the Terriers a first half lead which they worked hard to defend in the second half.
It was his only chance of the game and he took it with aplomb.
Danny Schofield had earlier had two chances to score in an even start to the match while Ryan William tested Martyn Margetson early on after being put through by Theodore Whitmore.
Margetson was to be much busier as the match went on.
Dangerman Gary Alexander was guilty of several wasted chances for the Tigers, once shooting wide from 5 yards with Margetson to beat.
Theodore Whitmore could and should have had a goal as well, with a wasted free header from 6 yards out after 73 minutes following a first half chance where he dribbled the ball into the Town area but didn't trouble Martyn Margetson with a wayward shot.
Lawrie Dudfield could have equalised Andy Booth's opener in the first half just two minute from half time, which would have made things interesting, but he could only blast Margetson's spilled catch from Andy Holt's chance back to the Welsh 'keeper - much to his relief.
Dudfield went on to miss a 56th minute chance from 5 yards out after neatly sidestepping Kevin Gray.
Town's luck was surely with them at Boothferry Park, as the second half Hull barrage proved, but some resolute defending, and a huge slice of luck saw them through to the next stage of the LDV Vans Trophy.
Town will now proceed to the Northern Area Final of the LDV Vans Trophy where they will play either Oldham Athletic or Blackpool and stand two games away from a trip to Cardiff's Millennium Stadium.
Hull (4-4-2): Musselwhite, Petty, Holt, Whittle, Goodison, Williams, Sneekes (Greaves 68), Whitmore, Beresford, Dudfield (Johnsson 68), Alexander
Subs not used: Glennon, Matthews, Mohan
Town (4-4-2): Margetson, Heary, Evans, Gray, Moses, Hay (Holland 77), Irons, Mattis, Schofield, Knight (Wijnhard 79), Booth
Subs not used: P. Senior, Clarke, Macari
Scorer: Booth 38
Referee: George Cain (Bootle)