Muralitharan bowled five maiden overs in his nine-over stint at Bogra District Stadium to leave Bangladesh, who won the toss, on 94-2 at lunch.
Bangladesh openers Javed Omar and Nafees Iqbal had compiled an impressive opening partnership of 52 runs before Muralitharan trapped Iqbal lbw on 26 with the fourth ball of his first over to break the partnership.
He later removed Javed Omar (35) also lbw in his seventh over with the score at 85.
Captain Habibul Bashar was not out on 22 with Shahriar Nafees on three.
Sri Lanka earlier this month scored an eight-wicket win over Bangladesh in the first test at Chittagong.