The 23-year-old opener was unbeaten on 130 for his fifth Test century as West Indies reached 314-7 at lunch, 33 ahead of Pakistan's first-innings total of 281 all out.
Brathwaite, batting with resolve, found an able ally in Devendra Bishoo as they combined to add 51 invaluable runs for an unbroken eighth-wicket stand.
Bishoo was 22 not out at the break, having supported his more established partner to leave Pakistan's spin-cum-pace attack struggling for wickets on a still unresponsive pitch.
In the same over, Bishoo was given out off a sweep shot but he successfully reviewed English umpire Michael Gough's decision as the ball had hit fielder Azhar Ali's helmet before he took the catch.
Resuming at 244-6, the West Indies were looking for a good lead to put pressure on Pakistan for the first time in the series as they fight to avoid a 3-0 whitewash.
Pakistan took the second new ball with the total on 251 and Amir struck instantly, clean-bowling Holder with a sharp incoming delivery for 16.
Amir has figures of 3-71.
Pakistan lead the three-match series 2-0 after winning the first Test by 56 runs in Dubai and second by 133 runs in Abu Dhabi.
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