Asif had Taylor caught off an uppish sweep for 82 before New Zealand reached 222-4 at lunch on the fourth day, still needing 106 runs to avoid an innings defeat.
At the break, Henry Nicholls, who successfully reviewed a second-ball leg-before decision on nought, was unbeaten on 31 with BJ Watling 16 not out.
New Zealand lead the three-match series 1-0 after winning the first Test by four runs in Abu Dhabi.