Goals from Max Wright and Michael Coulson made sure of a 2-0 victory, that eventually, Boro richly deserved.
Boro made a number of changes in the frozen weather, with Tom Morgan starting in between the sticks, while Danny Stimpson, Nathan Valentine, Jimmy Beadle and James Cadman also grabbed shirts. Lewis Sugden, the new signing from across the pond, came in for his debut.
Little happened of note in a quiet first half, with both sides struggling to cope with hugely challenging conditions.
Marske possibly battled on the better in the early stages, getting on the ball and playing it around, while Boro chased around, struggling to keep their feet.
But as the game went on, Boro began to warm up and their opponents gradually froze.
The hosts looked to strike neatly on the counter and could well have been ahead by the interval.
Stimpson's far-post header was a shade wide of the upright, then Valentine was nudged legally off the ball as he was about to pull the trigger.
Those chances sandwiched Marske's only first-half effort of worth, when Morgan palmed away a drive from former Whitby man Curtis Round.
Beadle had an amazing opportunity after the dancing feet of Nathan Curtis picked him out a handful of yards from goal, but he failed to get his shot away and Marske cleared.
Boro finished the half with two long-range efforts, a drive from Valentine and a Coulson free-kick, with both grazing the upright.
The new period saw a sharper beginning, with Marske drawing a couple of smart saves from Morgan in the Boro goal.
Round again found himself in a good position, as the Boro defence parted in front of him, but his strike from the edge of the box was truly awful and sliced well wide of the target.
At the other end, Boro's Coulson wriggled into some space of his own, but his shot was deflected over the bar.
Boro had a moment of concern just before the hour, when Mikey Roberts shrugged off Stimpson and rounded Morgan, but Sam Hewitt got back to make a crunching challenge, which forced the ball to safety.
James Fairley was the next to threaten as he was picked out on the angle of the box, but he bent well wide.
With Boro having introduced Max Wright at the break, they continued to shuffle, with Ross Daly and James Walshaw being thrown on in the search of a needed spark.
And the Boro breakthrough eventually came, but this pre-Christmas present was wrapped up by Marske keeper Lewis McDonald.
He gave the ball straight to Wright, who arrowed into the box and picked out the bottom corner.
Walshaw was unlucky not to double the lead seconds later after an interchange with Wright. He burst in on McDonald, but his shot grazed the upright.
Morgan earned more of his corn with a good block to deny sub Peter Bulmer, before Boro wrapped up the victory.
It was Coulson who collected on the half-way line, he darted past a number of players and drilled into the back of the net.
This was the perfect way for Boro to sign off, as they comfortably saw out the final moments and made sure they progressed through to the next round.