After bowling out Pakistan for 628 just before lunch, Bangladesh was 142 without loss in its second innings at the second interval.
Tamim, who hit centuries in his previous two Tests against Zimbabwe in November, was not out on 85, and Kayes unbeaten on 56.
Hitting nine fours and three sixes in his 98-ball knock, Tamim helped to reduce Pakistan's significant advantage on the flat pitch at Sheikh Abu Naser Stadium.
Kayes, who scored 51 in the first innings, was solid as he sent seven deliveries to the boundary, and hit one six in his 89 balls.
On the unthreatening pitch, the Bangladeshi openers made Pakistan toil in the whole second session, scoring runs at a rapid pace.
Making a mockery of the Pakistani bowling, they scored at more than 4 1/2 runs per over.
Earlier, Pakistan established a 296-run first-innings lead, moving from 537-5 overnight to 628.
Left-arm spinner Taijul Islam took three of the five wickets to finish the innings with 6-163 off 46.4 overs.
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