The two added 155 runs for the fifth wicket.
At close, Kerala were 168 for one with opener A M Hegde (74) and V A Jagadeesh (92) at the crease.
Kerala now lead by 146 runs with nine wickets and a day left.
Earlier, resuming on 161 for four, Assam's Jadhav dominated the proceedings, though both were unfortunate to have missed centuries.
Jadhav faced 210 balls and hit nine fours and two sixes while completing 5,000 runs in Ranji Trophy when he reached 97.
Singh hit eight boundaries and two sixes while facing 181 balls.
Middle-order batsman R N Das contributed an invaluable 36 for Assam while for Kerala medium pacer Sony Cheruvathur 3/45 and Anthaf 3/63 shone with the ball.
Having conceded the first-innings lead, Kerala got off to a shocking start in their second innings, losing opener R V Gomez in the very first over from medium pacer Ahmed.
However, veteran Jagadeesh, who joined opener Hegde, went after the bowling and brought the side back into the game.
Brief scores:
Kerala: 264 and 168/1 (Hegde batting 74, Jagadeesh batting 92, Ahmed 1/20)
Assam 286 (D S Jadhav 98, Tarjinder Singh 92, R N Das 36, Sony Cheruvathur 3/45, Anthaf 3/63).
