Sri Lanka went to tea on the fourth day on 195-3 in their second innings, an overall lead of 356 runs with seven wickets in hand on a sluggish pitch at the Zahur Ahmed Chowdhury stadium.
Left-handed Sangakkara, who slammed a monumental 319 in the first innings, held the faltering innings together with an unbeaten 90 at the break.
Sangakkara is now just 10 runs away from joining Englishman Graham Gooch as the only batsmen to score a triple-century and century in the same Test.
Off-spinner Mohammad Mahmudullah dismissed both openers, Dimuth Karunaratne and Kaushal Silva, in his first three overs before lunch to leave the tourists tottering at 49-2.
Left-handed Karunaratne was snapped up smartly in the slips by Nasir Hossain for 15, while Silva was trapped leg-before for 29.
Shakib Al Hasan then claimed Mahela Jayawardene leg-before for 11 early in the post-lunch session as Sri Lanka slipped to 78-3.
But Dinesh Chandimal helped Sangakkara add 117 for the unbroken fourth wicket as he went to tea on 49 not out.
Earlier, Bangladesh were bowled out for 426 in their first knock within 25 minutes of the start in reply to Sri Lanka's 587.
Mendis dismissed Mahmudullah with the morning's fourth delivery, forcing a catch to short-leg before the batsman had added to his overnight score of 30.
Al-Amin Hossain was bowled by Mendis for nine, leaving last man Abdur Razzak unbeaten on 11.
Off-spinner Dilruwan Perera supported Mendis with three wickets, while seamer Suranga Lakmal took one.
The tourists are looking to sweep the short series after winning the first Test by an innings and 248 runs in Dhaka last week.
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