Taylor's 104 for his 12th Test century enabled New Zealand to declare their second innings at 250-9 and push for a series-levelling win on the fifth and final day.
At the lunch break Shan Masood (11) and Azhar Ali (13) were at the crease as Pakistan need 229 more runs to win in 58 overs or bat out the last two sessions for a draw.
Pakistan lead the three-Test series 1-0 after winning the first Test by 248 runs in Abu Dhabi last week.
Earlier Taylor anchored the New Zealand innings.
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Taylor, who had just 31 runs in the three previous innings of the series, batted with authority during his 200-minute stay at the crease studded with 12 boundaries.
New Zealand, resuming at 167-6, looked for quick runs which came once Taylor reached his hundred with a single off seamer Ehsan Adil. It took him 129 balls to reach the three-figure mark.
Leg-spinner Yasir Shah, who finished with his first five-wicket haul for figures of 5-79, removed Taylor and Mark Craig (34) in the same over.
Southee hit Zulfiqar Babar (4-96) for three towering sixes before he was caught in the covers and Brendon McCullum declared.
The third and final Test starts in Sharjah from November 26.