The tourists, replying to Bangladesh's first innings total of 232, took their overnight score of 60-0 to 139-1 by lunch on the second day at the Sher-e-Bangla stadium.
Left-handed Karunaratne made 53 before he was snapped up at mid-off when he mistimed a drive off left-arm spinner Shakib Al Hasan.
Silva returned unbeaten on 66 and Kumar Sangakkara was on 11, with Sri Lanka only 93 runs behind with nine wickets in hand on what remains a good batting pitch.
Silva, who lofted Sohag Gazi for a six to long-off, has also hit hit seven fours in his fourth Test half-century.
But Silva was lucky to still be at the crease, having been dropped by Bangladesh captain Mushfiqur Rahim behind the stumps and again by Shamsur Rahman.
Silva had another reprieve when he was caught behind off a no-ball by seamer Al-Amin Hossain.
The two Test matches will be followed by two Twenty20 matches and three one-day internationals.