Vertonghen, a £9m summer arrival from Ajax, was one of just three players maintained by Andre Villas-Boas from the side that defeated Queens Park Rangers on Sunday.
He ended 37 minutes of stubborn Carlisle resistance with a first-half header, before Andros Townsend made sure of the result with a wonderful solo effort in the second period.
Townsend doubled Spurs' advantage with a curling low shot after dribbling past two Carlisle defenders with a run from the halfway line.
Villas-Boas had the luxury of substituting Vertonghen and Dempsey, with an eye on this weekend's clash with Manchester United, and it was one of the replacements who set up Spurs' third.
Kyle Walker made an overlapping run from left-back before crossing for Sigurdsson to head in at the back post.