The Seasiders ran out 4-1 winners against a youthful York City side, with only Mackenzie Heaney, Jonny Haase and Harrison Hopper with real first-team experience this season for the Minstermen, writes Paul Connolly.
Going into the game, Whitby made a number of changes themselves, with the likes of Coleby Shepherd and Jake Hackett starting, while Lee Bullock opted to take to the field for the first time this season, partnering Marcus Giles up front.
Despite the changes ringing though, Haslam believed it was a good showing from his side.
"It was a comfortable win in the end," the joint-Blues manager said.
"It must be said that it was a young York City side but they equipped themselves well.
"They tried to play the game, they were fit, they got through the difficult conditions but I think we overpowered them, showed a little bit of class and won 4-1."
Whitby opened the scoring just seven minutes into the game, though had York City's keeper Mason Campbell to thank. Josh MacDonald's inswinging corner caught Campbell out, who fumbled the ball in the six yard box and into his own net.
The Blues had chances to extend that lead across the first half too - Bullock's glancing header from Hackett's delivery was deflected wide, while the likes of Nathan Dyer and Marcus Giles also tested Campbell.
York had chances to level matters too, most notably with Shane Bland pushing Sam Fielding's near-post free-kick wide, before Jake Hackett headed off the line from the resulting corner.
Going into the break, Whitby were forced into a triple change, with injuries causing issue for the Seasiders.
"We're down to the bare bones. We've got a few niggles and knocks and had to play Lee [Bullock] tonight who's had a decent game.
"We had three lads who needed to come off at half-time. Adam Gell with a reaction to his booster jab, Jonny Burn with an old injury flaring up and Marcus Giles with a tight calf, so we made all three subs at the break. We're thankful we've come through unscathed from that point."
The second for Whitby came shortly after the break. The ball falling nicely for Jacob Hazel in the penalty area to finish calmly inside the near post.
Shortly after Whitby's talisman had the chance to make it three from the spot after Brad Fewster was bundled down inside the area, though could only fire over the bar, with Whitby's woeful spot-kick record this season continuing.
York then found a way back into the game through the former Whitby man Heaney, who fired a low free-kick across Shane Bland and into the bottom corner - a carbon copy of the set-piece he scored for the Minstermen in a pre-season clash, back in July.
Whitby, however, were looking to set up a mouthwatering clash at Scarborough Athletic in the Quarter Finals and got the job done with two quality strikes within a minute of each other.
Coleby Shepherd was involved in both efforts - netting the first with a low, twenty-yard drive after receiving the ball from Hawkins on the left-hand edge of the box, before turning provider when he weighted a lovely ball over the top for Priestley Griffiths, who got between the defender and goalkeeper to chip the ball over the onrushing gloveman.
And Shepherd's contributions were particularly pleasing for Haslam, who had asked the young winger to have a go for goal more often at the break.
"I set the challenge for Coleby at half-time to get his shots off," he added.
"He's done what I've asked of him and fair play. It was a great finish and it epitomised his performance.
"He was top drawer and the best player on the pitch by a country mile."