The 25-year-old, playing his first Test since 2012, kept Pakistan at bay with an unbeaten 74 and found an able ally in Jonny Bairstow (37 not out) as the two put on an invaluable 83 for the unbroken fifth-wicket stand at Sharjah Stadium.
That leaves England just 12 behind Pakistan's first-innings total of 234 with six wickets intact as they hope to gain a decisive lead in their bid to level the three-match series.
Taylor kept England in the hunt as he survived a keen tussle between bat and ball, with Pakistan's spin duo of Yasir Shah and Zulfiqar Babar threatening to take wickets.
But Taylor, playing his first Test since his two debut Tests against South Africa three years ago, batted with resolve, using his feet and sweeping to blunt the spin.
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He pushed paceman Wahab Riaz for a single to reach his first half century off 100 balls. He has so far hit six boundaries during his 141-ball fight.
Taylor added 42 for the fourth wicket with Ian Bell (40) to steady the innings as Pakistan attacked through both pace and spin.
For Pakistan Shah took 2-79 while Babar went wicket-less in his 25 overs and even a second new ball at 181-4 failed to dislodge the Taylor-Bairstow partnership.
Bell, who survived a confident caught behind appeal off Babar, was finally stumped off Shah. He hit two boundaries and a six during his three hour 37 minute innings.
England had added three runs to their lunch score of 87-1 when Cook fell to a soft dismissal as he pushed a Shah delivery straight into the hands of short leg fielder Azhar Ali.