The Robins romped their way into the second round with goals from Pell, Barthram, Waters and Holman putting Crewe to the sword.
Ryan Lowe brought one back for the Railwaymen, but it’s Johnson’s men who progress to the next round, setting up a tie away at Sutton United.
Crewe had the ball in the net within the first two minutes, but Udoh was adjudged to have fouled Rob Dickie in the box.
A subtle dummy from the returning Holman created space for Waters, however the former’s effort was off target.
Cheltenham’s pressure was rising and the deadlock was broken when Wright stole the ball on the halfway line, fed Holman, whose effort was blocked, with Harry Pell there to prod home for his fourth of the season.
It was 2-0 just 20 minutes later. Rob Dickie was allowed to waltz forward 50 yards with the ball, he slotted a perfectly weighted pass through for Jack Barthram to poke home his first ever goal for the Robins.
Cheltenham had controlled the whole half. Crewe were simply unable to cope with the physicality and intensity of the Robins side.
Crewe brought on Ryan Lowe at half-time to try and test the Robins back five with his strength, and the Railwaymen nearly brought one back, but Guthrie’s effort was wild.
However, the important third goal was scored by the Robins. O’Shaughnessy’s long punt into the box found Waters via Holman, and the ex-Crewe man was on hand to make it 3-0.
It was then roles reversed for another Cheltenham goal. Waters threaded one down the line for Holman, who needed no invitation to cut in and smash it into the far corner.
Crewe brought one back just minutes later through Lowe after some neat play from Saunders down the left, cutting back for the striker to finish.
It was so nearly five after the industrious Waters and Morgan-Smith, brought on for Holman, combined. Waters’ pass across goal was dinked cutely onto the top of the bar by a stretching Morgan-Smith.
Cheltenham, still hungry for more, had further chances saved through Danny Wright and Amari Morgan-Smith.
Jubilation at full-time for the Robins supporters who were magnificent throughout. It was exhibition football at times from Johnson’s boys, and they were much deserved winners.
Cheltenham Town FC: Griffiths; Barthram Downes Parslow Dickie O’Shaughnessy; Pell Dayton (Munns 58’); Waters (Rowe 81’) Holman (Morgan-Smith 67’) Wright
Crewe Alexandra FC: Garratt; Ng Davis Nugent Guthrie; Cooper Hollands (Wintle 67’) Jones Ainley Kiwomya (Lowe 45’); Udoh