Paul Ince's men made most of the running in the first half but Pool goalkeeper Matt Gilks was forced into impressive saves from Junior Stanislas and Ross Wallace, the former aided by the base of his post.
Sylvestre gave Pool the lead in their 1-1 draw with Nottingham Forest last weekend and he broke the deadlock here with a superb long-range strike after 58 minutes.
Ben Mee went back into the Burnley team at left-back, Sean Dyche's only change from last weekend's 3-0 win over Bristol City.
Veteran Blackpool midfielder Barry Ferguson returned from a one-match ban as Isaiah Osbourne began a suspension of his own, while Gary Taylor-Fletcher was named on the bench for the first time since suffering a hamstring injury early last month.
Blackpool's on-loan Aston Villa forward Nathan Delfouneso spurned a seventh-minute opportunity to open the scoring, scuffing a shot into the grateful arms of Burnley goalkeeper Lee Grant.
Seasiders full-back Kirk Broadfoot raided forward to flash a shot past the far post from the right, but Burnley almost took the lead with a quarter of an hour played as Stanislas shot at the end of a driving run to be thwarted by a combination of Gilks and the inside of the upright.
Back came Blackpool and Grant required two attempts to gather when Tom Ince blasted into the turf from Delfouneso's knockdown.
Matt Phillips blasted over three times from the edge of the box as the hosts continued to ask questions but Burnley once again came closest to breaking the first-half deadlock three minutes before the interval.
Ross Wallace unleashed a swirling, dipping effort that Gilks did superbly well to touch beyond the crossbar.
After a quiet start to the second half, Sylvestre brought Bloomfield Road to its feet by going one better than Wallace.
Little appeared to be on as the French midfielder edged in from the left and curled a wonderful strike into the top corner.
Burnley almost hit back in the 67th minute when Gilks flapped Wallace's cross partially away from the goalmouth but Chris Basham blocked Stanislas' follow-up shot.
Dyche's men piled forward in vain and Blackpool would have been able to breath easier had Sylvestre not headed wide after Ince's deflected effort came back off the bar.