When umpires called for the end of the first session, India's score read 269 for five with Mohammad Kaif and Uday Kaul batting on 24 and 13, respectively.
West Indies were bowled out for 406 in their first innings.
Unheralded Kerala opener Jagadeesh, who showed good temperament for his undefeated overnight score of 79, added only seven runs to his tally before being dismissed.
Once Jagadeesh was packed off, Nayar (89) had to take a long walk back to the pavillion after he was trapped LBW by left-arm spinner Nikita Miller.
Right-arm fast Barbadian Miguel Cummins was the first to provide the breakthrough in the morning as he bagged the prized wicket of Jagadeesh before Miller took Nayar's wicket.