Mendis completed his century on the score of 171 by dismissing the full toss ball of Bangladesh captain Nazmul Hossain Shanto.
Mehdi gave some relief to Bangladesh by dismissing Prabhat Jayasuriya and Lahiru Kumara, who scored 25 runs, on consecutive balls.
Had Liton Das not dropped the catch of Mendis at the score of 133, Mehidy could have taken his fifth wicket.
Mendis saved the last man Rajitha by adding 52 runs for the last wicket and increased the problems for Bangladesh.
Mendis hit three sixes in one over of Taijul Islam, taking his score to 150 runs and taking the lead beyond 500 runs.
Taijul finally wrapped up Sri Lanka's innings by dismissing Mendis.
Sri Lanka has never lost a Test series to Bangladesh.
The second and last match of the series will be played in Chittagong from March 30.