Kevin Ellison's debut goal earned the Bantams their first win in seven to ease the pressure on Taylor.
Ellison, who joined City on loan from Rotherham in midweek, arrived at the far post to apply the finish in the 69th minute and was promptly booked for removing his shirt in celebration.
Taylor said: "It was a very important three points. You saw the difference in the two teams in the first half when they looked very confident and we looked nervous.
"In the second half we carried on working hard, the crowd got behind us even more and we scored at a very important time, which gave us a big boost."