Junior Mondal fired home on 24 minutes to give a dominant Town the advantage, just eight minutes before the home side conceded a second, writes Liam Ryder.
Summer signing Jerome Greaves then tapped him his third competitive goal for the club following some fantastic work from Soni Fergus who had broken forwards from right-back.
It was no less than Nathan Haslam's side deserved having had much the better of the first 45, with Mondal also calling Jody Barford into action not long before his goal.
Their dominance continued after the break, Storr making it three before he added Town’s fourth, and Haswell grabbed a double of his own.
Haslam opted to shuffle his pack minimally with Fergus, Greaves and Jerome Greaves being the only different starters in the side that defeated Ilkeston Town 4-2 in Town's previous outing.
Both sides took time to settle into the game with Goole's long balls over the top causing some early issues for the Blues, while the away side opted for a slower, more relaxed approach in the early stages.
The dangerous Mondal was keeping Goole's backline occupied, and he almost fired home in the 17th minute, only to be denied by Barford.
With Whitby starting to get on the front foot, it seemed only a matter of time before the deadlock was broken - and a first goal duly arrived, courtesy of Mondal.
Alfie Doherty slipped the ball through to the attacker who then drilled a low effort into the far corner of the net.
Having conceded, Goole's woes were compounded even more when they conceded a second as Greaves was the beneficiary of some unselfish play from Fergus.
A string of attacking changes in the second period indicated that Town were in no mood to take their foot off the gas with Storr and Sam Spenceley introduced early on.
And it didn't take Storr long to get his name on the scoresheet as he latched onto Doherty's through ball to smash a strike home beyond Barford to all but make sure of Whitby's place in the next round.
With 18 minutes to play, the crowd of 838 at the Victoria Pleasure Grounds were treated to another strike from Whitby-based Storr as he controlled the ball well from Adam Gell's pass before bending a strike beyond Barford to delight the strong travelling contingent in attendance.
The onslaught continued as both Daniel Rowe and Jacob Gratton struck the woodwork, but there was still time for another two goals from the Blues who showed no mercy against their NCE League opponents.
It wasn't until the final 10 minutes that the hosts mustered an attempt on goal, and Shane Bland didn't have to worry about a thing as Liam Flanagan's curling effort from distance was high and not very handsome.
Just seconds later it was five. Storr strode forward and slid the ball through to Haswell who found the net from a tight angle.
And with five minutes left, Haswell used some subtlety to dink the ball over the on-rushing Barford to round off a wonderful night for Haslam's men.