Spurs took the lead in the 37th minute when Vertonghen headed in Sigurdsson's set-piece delivery, the Belgian registering his first goal for the club.
Townsend then doubled the advantage with a decent solo effort eight minutes after the restart before Sigurdsson, also opening his account for the visitors, wrapped the scoring up by nodding in Kyle Walker's cross a minute from time.
Tottenham boss Andre Villas-Boas would have been relieved to avoid a potential banana skin at Brunton Park as his team - showing eight changes from the 2-1 Barclays Premier League win over QPR on Sunday - dealt with what Carlisle were able to throw at them and did enough at the other end to complete the job.
The debate over Spurs' goalkeepers will continue after Villas-Boas' selection tonight, with the Portuguese opting to play Carlo Cudicini and name Heurelho Gomes as his back-up on the bench, with Hugo Lloris not featuring in the squad.
Overall, though, it seems the pressure on the manager is continuing to ease, with this result meaning that ahead of Saturday's league match at Manchester United, the London outfit are undefeated in the six games since their loss to Newcastle on the opening day of the top-flight campaign, and have won three of their last four in all competitions.
Carlisle went into the contest with some form themselves having put together a five-match unbeaten run, and showed early on that they were not afraid to run at their opponents.
Tottenham were the ones carving out the real opportunities, though, with United goalkeeper Mark Gillespie tipping over an effort from Yago Falque in the ninth minute.
Spurs forward Clint Dempsey should have done better in the 25th minute when he nodded over from Townsend's cross, and soon after the United States international saw an attempt cleared off the line by defender Danny Livesey.
The hosts continued to press when they could, but it was Tottenham who made the breakthrough eight minutes before the interval as Vertonghen nodded in Sigurdsson's free-kick from the right.
Carlisle looked for a swift response and Danny Cadamarteri's strike was deflected over via the head of Michael Dawson, who was appearing for the first time under Villas-Boas.
A nice move then saw United's Kallum Higginbotham teed up, but his horribly skewed finish summed up the home side's lack of cutting edge.
It was apparent once more shortly after the break as Cadamarteri dragged an effort wide before Townsend demonstrated some Premier League potency by drilling in Spurs' second, the midfielder bursting forward and shooting low past Gillespie.
Gillespie then did well to keep out a Ryan Mason header before pulling off a further double save, pushing a header from the same player against the bar and then denying Sigurdsson.
Carlisle could muster little in reply and Spurs added a third in the 89th minute as Sigurdsson headed in from substitute Walker's delivery.| Carlisle United | Team Statistics | Tottenham Hotspur |
| 0 | Goals | 3 |
| 0 | 1st Half Goals | 1 |
| 4 | Shots on Target | 8 |
| 4 | Shots off Target | 7 |
| 0 | Blocked Shots | 0 |
| 10 | Corners | 7 |
| 14 | Fouls | 12 |
| 2 | Offsides | 0 |
| 0 | Yellow Cards | 0 |
| 0 | Red Cards | 0 |
| 37.3 | Possession | 62.7 |
