After a tight opening to the tie Rangers forced the visiting goalkeeper into action with Louis Briscoe having his initial effort save.
Jack Sherratt was next to prore for the Boro and had two glorious chances to take the lead from within the six-yard box.
Next to have a go at goal was Andrew Haworth firing at the goalkeeper before his side wasted a two on one opportunity.
The deadlock was eventually broken on the 53rd minute with Biscoe finding the top corner with his effort from inside the box.
Despite the hosts looking in control Sports almost had a fortunate leveller on the 67th minute when substitute Jack Storer’s cross come shot rattled the bar.
However, Boro were not to be denied a place in the next round as skipper Luke Jones secured their passage with a headed second goal before Evan Garnett netted a third.