Having been pegged back by visiting Ossett Town at the end of the first half, Boro dug in and managed to edge their way to a 2-1 success, which booked their path into the last 16.
It was by no means a classic contest at a freezing Flamingo Land Stadium, but Boro showed flashes of brilliance and that proved to be enough.
Little or nothing happened in the opening moments of the game, apart from the naughty challenging from Ossett, the worst of which from Steve Ridley was punished with an early yellow card.
Boro began to click through the gears, and the dancing feet of Max Wright continued to trouble former Athletic man Ridley.
Max Wright put the ball on a plate for new signing James Walshaw, but the striker failed to set his feet and the chance was gone.
Then the winger picked out the head of Ross Daly, but his header was deflected over the bar.
Ossett were eventually undone by their aggressive style of play, as Sam Wright went in with another late one on Max Wright.
He was carded for this, but then castigated further in the following seconds. Dave Merris whipped in a glorious free-kick and Sam Hewitt prodded home.
Just as it looked as though Boro would canter into the break, their stride was clipped by an Ossett leveller.
Returning striker Emile Sinclair broke free of the Boro defence, before cutting back and scuffing a shot over Tommy Taylor and James Knowles on the line.
After returning from the warmth of the changing room, Boro looked revitalised when the second half began, especially after the introduction of Billy Logan in place of an injured Walshaw.
Logan managed to ghost onto a Michael Coulson cut back with just a few minutes on the clock, but his effort was blocked by Edward Hall in the Ossett goal.
This was a trait that continued as the half progressed, with Boro huffing and puffing in attack, but failing to blow Ossett's house down.
The West Yorkshire side seemed content to sit in and then attack on the counter, using the pace of Sinclair.
He streaked clear of the Boro backline on one of his forrays and drew a save from Taylor. The rebound fell to Danny South, but his effort was blocked on the line by the covering Danny Stimpson.
Boss Steve Kittrick made a throw of the dice on the hour, bringing off the battling Jamie Price and throwing on Jimmy Beadle.
Typically Beadle made the desired impact, finding his way onto a bobbling ball in the box just minutes after coming onto the pitch, and crashing it home to restore Boro's lead.
Beadle then had a drive from the edge of the box held by the keeper as the hosts looked to put the game to bed.
Sinclair had the chance to pull the sheet from under them in the final moments, when marauding onto a ball into the box. The striker beat the offside trap, but then a rush of blood led to him lifting over the bar and into the Shed from five yards.
Marshalled superbly by James Knowles, Boro saw out the last few minutes to make sure they returned to winning ways.