Kamindu followed a first-innings 102 with his career-best 164 and de Silva, who also scored 102 in the first innings, made 108 as Sri Lanka were dismissed for 418 in their second innings, setting an improbable 511-run victory target for Bangladesh.
The hosts then collapsed in reply and were reduced to 5-47 at stumps on Sunday's day three, after Fernando finished with 3-13.
Bangladesh are 464 runs short of securing an unlikely victory.
Kamindu and de Silva put on 202 as Sri Lanka recovered from 5-57 to post 280 before being all out in their first innings.
Fernando then claimed 4-48 to help the side bundle Bangladesh out for 188.
Fernando struck in the first over of Bangladesh's second innings when he had opener Mahmudul Hasan lbw for a duck before captain Najmul Hossain Shanto top-edged a wide delivery off pacer Kasun Rajitah to Dimuth Karunaratne at first slip.
Zakir Hasan looked positive but he too poked a delivery outside when on 19 and left Bangladesh at 3-36.
The situation was compounded when Fernando struck twice in consecutive deliveries, dismissing Shahadat Hossain and Liton Das for a duck.
Resuming at 5-119, Sri Lanka lost nightwatchman Fernando in the third over of the day but de Silva and Kamindu then kept the home side frustrated throughout the session.
After raising his half-century off 82 balls, de Silva paced the innings, smashing offspinner Mehidy Hasan for consecutive sixes.
He raised his 12th century off 164 balls with a single off Mehidy but a short time later the bowler, who took 4-74, got the better of de Silva to end his 173-run partnership with Kamindu.
De Silva, who became the sixth Sri Lanka batter to hit centuries in both innings of a Test, struck 108 off 179 balls in a stint laced with nine fours and two sixes.
Kamindu, whose knock featured 16 fours and six sixes from 237 balls, reached his second hundred off 171 balls and joined the elite list of the eight batters to achieve the feat by pushing Shanto for a single.
Getting more aggressive, he then raced to 150 off 228 balls, smashing Taijul for six over long-on but one extravagant shot too many saw him top-edge Taijul to Mehidy at midwicket, which ended Sri Lanka's innings.
