Masood was bowled by off-spinner Mark Craig for 12 but fellow opener Mohammad Hafeez (47 not out) and Azhar Ali, unbeaten on 20, steadied the innings.
Pakistan, seeking their second straight series victory after routing Australia 2-0 this month, got a kick-start by winning a crucial toss. Skipper Misbah-ul Haq immediately opted to bat on a dry, flat and grassless Sharjah stadium pitch.
Hafeez, who missed the second Test with a hamstring injury, looked solid and struck six boundaries and a six.
Pakistan made two changes from the Dubai Test line-up, dropping out-of-form opener Taufeeq Umar and Ehsan Adil, who suffered a side strain. They brought in Hafeez and paceman Mohammad Talha.
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New Zealand entered the Test with three spinners after recalling veteran left-armer Daniel Vettori, playing his first Test for 28 months.
He became his country's most capped player with his 112th Test -- 113th overall since he played one Test for an ICC World XI in 2005) -- beating Stephen Fleming who played 111.
Pakistan lead the three-match series 1-0, after winning the first Test in Abu Dhabi and drawing the second in Dubai.