The hosts, facing a monumental Sri Lankan first innings total of 587, fought splendidly to reach 409-8 by stumps on the third day at the Zahur Ahmed Chowdhury stadium.
It was a remarkable comeback by the bottom-placed Test side, who had been thrashed by an innings and 248 runs in the first Test in Dhaka last week -- suffering their 14th defeat in 15 matches against Sri Lanka with one draw.
Rahman and Kayes, who came together when Tamim Iqbal was dismissed without a run on the board on Wednesday afternoon, put on Bangladesh's highest second-wicket stand of 232.
Rahman scored 106 in only his second Test and left-handed Kayes celebrated his return to the Test side after two years with 115 to lift Bangladesh to 232-1 soon after lunch.
The duo surpassed their side's previous best partnership for the second wicket of 200 by Tamim Iqbal and Junaid Siddiqui against India in Dhaka in 2010, but they gave away their wickets with rash strokes.
Sri Lanka sliced through the middle order before Shakib Al Hasan retrieved the situation with a half-century and Nasir Hossain chipped in with 42.
Mendis dismissed Nasir and Sohag Gazi off successive deliveries, but Al-Amin Hossain prevented a hat-trick by defending the next ball.
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