After a good start to the tie for Rangers, it was the visitors who took a 16th minute lead when Glenn Steel headed his side ahead.
Despite the opener the Boro kept pushing for an equaliser but were denied by the City defence led by Ibrahim Bakare.
Another set piece saw Lancaster double their lead with Sam Bailey’s free-kick met by striker Matthew Blinkhorn.
Rangers once again responded but were met by a stubborn defensive line and saw one effort fly over the bar.
However, they were given hope when the visitors were reduced to 10 men after Charlie Bailey saw red for a second bookable offence.
Rangers probed again after the break and forced goalkeeper Samuel Ashton into action before seeing another effort fly over.
However, it was Lancaster who sealed the points on 63 minutes when Bradley Carsley beat goalkeeper Lewis King to a high ball before gratefully scoring.