The on-loan Chelsea man struck after 48 minutes, racing onto Luke McCormick's ball before emphatically firing into the top corner from 12 yards.
Palace had their chances, with defender Leon Cort shooting wide from 12 yards in the first half.
And after Sinclair's goal, Shefki Kuqi twice went close but was denied by Pilgrims' keeper Luke McCormick.
Palace boss Peter Taylor:
"We have got seven matches to go and play four of the teams that are above us.
"Three of them are at home and one of them's away, so it's still a little bit in our hands.
"But we cannot afford to slip up again and we could have done with something today."
Plymouth: McCormick, Connolly, Seip, Doumbe, Sawyer, Gosling (Hodges 83), Norris, Nalis, Sinclair, Ebanks-Blake (Fallon 68), Hayles.
Subs Not Used: Gallen, Summerfield, Djordjic.
Goals: Sinclair 48.
Crystal Palace: Kiraly, Butterfield, Hudson, Cort, Borrowdale, Soares (Kuqi 71), Watson, Fletcher, Hughes (Freedman 59), Scowcroft, Morrison (Grabban 71).
Subs Not Used: Speroni, Kennedy.
Ref: R Booth (Nottinghamshire).