Sylvestre volleyed the Tangerines to safety and sealed a third straight win for Paul Ince's team at Bloomfield Road.
The game looked like ending in a draw after Derby substitute Ben Davies cancelled out Gary Taylor-Fletcher's opener for the hosts.
Manager's son Thomas Ince spurned a chance to give his side a two-goal lead when he struck his 33rd-minute penalty against a post.
Taylor-Fletcher tapped in his fifth goal of the season, all of them coming at Bloomfield Road, from close range on 18 minutes after Ince picked him out.
The hosts were handed the chance to give themselves a two-goal cushion when referee Gavin Ward pointed to the penalty spot following Craig Bryson's challenge on Kirk Broadfoot.
The Derby midfielder appeared to win the ball but the Rams will feel justice was served as Ince's kick struck the upright.
Adam Legzdins made his first save of note at the beginning of the second half, helping Stephen Crainey's free-kick over the crossbar. The visitors survived another scare in the 50th minute when John Brayford cleared Taylor-Fletcher's shot off the line.
A series of corners resulted in Derby equalising after 65 minutes. Davies had only been on the pitch three minutes after replacing Conor Sammon but his impact was immediate, volleying home Chris Martin's cross.
Bryson then went close with a curling effort from the edge of the box.
Legzdins had to save on his goalline from Nathan Delfouneso with five minutes remaining before Davies made two clearances off the line which looked like ensuring Derby would earn a point.
However, Sylvestre let fly with a volley from the edge of the box to condemn Nigel Clough's team to a 13th away league defeat.