The tourists, who won the first Test by an innings and 248 runs in Dhaka last week, scored 587 in their first-innings and Bangladesh replied with 86 for one by stumps on the second day.
Left-handed Sangakkara's maiden triple century was his ninth 200-plus knock, leaving him in second place behind Australian legend Don Bradman who had 12 scores of 200 or more.
Sangakkara, a 36-year-old veteran playing his 122nd Test, also became the ninth Test batsmen to reach the 11,000-run mark.
He now has 11,046 runs in the all-time list headed by India's Sachin Tendulkar (15,921).
"As a batsman, you always look at big scores," said Sangakkara. "There are 100s, 200s, 300s and Brian Lara has even got 400. Those are things you always try and chase when you have an opportunity.
"So I am very happy to have got into the 300-club at least."
Having started the day on 160, Sangakkara hit 32 boundaries and eight sixes before he was last man out before tea attempting to force the pace.
Bangladesh's reply began on the wrong foot when star opener Tamim Iqbal was bowled by Suranga Lakmal with the fourth ball of the innings.
But Shamsur Rahman (45 not out) and Imrul Kayes (36 not out) repaired the early damage by adding 86 for the unbroken second wicket.
Both batsmen were lucky to survive till close as Shamsur was dropped by Mahela Jayawardene on 36 and Nuwan Pradeep grassed a simple catch offered by Kayes just before stumps.
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