Spurs, who were without Gareth Bale because of a foot injury, started their season with a 1-0 win at Selhurst Palace thanks to a debut goal from record signing Roberto Soldado.
The Londoners wasted a series of chances to kill off the game in the second half, with Jermain Defoe and Gylfi Sigurdsson both guilty of dragging the ball wide when one-on-one with Eagles goalkeeper Julian Speroni.
Tottenham's profligacy almost cost them dearly as Hugo Lloris was forced to make two fantastic saves at the death to keep his team in the lead.
Having been part of the squad that came so close to pipping Arsenal to fourth last season, Vertonghen is aware that every goal this season will be precious and he hopes to see an improvement in the near future.
"We were very dominant, but we kept them alive and they had a big opportunity in the 89th minute," the Spurs defender said.
"We should have more of a killer instinct. If we did we could have scored two or three more, then we would have been able to relax a bit and make more substitutes.
"We were happy with the win. It would have been a better result if we had got two or three goals more."
Despite his concerns, Vertonghen has been impressed with how the four new faces on display performed on Sunday.
Soldado showed composure to slot home the winning penalty while Paulinho, Nacer Chadli and second-half substitute Etienne Capoue all showed the drive and creativity that will be required should Bale depart for Real Madrid.
Midfielder Mousa Dembele thinks Tottenham are now in a much better position to challenge Arsenal, who have not spent a penny this summer, for a top-four finish.
"We had quite a good season last year but at the end we didn't make what we aimed for," Dembele said.
"But this year we have learned from this and we can't make the same mistakes again.
"We have the quality (to make the top four). It's going to be very difficult because everyone has spent money, but we have bought some good players too.
"We have also been playing a year longer together. Everyone believes that we can do something this year. We have bought some good players. I can see already that we are playing well, but we can still play much better."
Of those new arrivals, Dembele knows Chadli the best from their time together with Belgium.
Chadli, signed from FC Twente for £7million this summer, played on the left wing at Selhurst, but he can also be deployed as a striker or as a number 10.
Dembele thinks the 24-year-old could prove to be a crucial signing for the club.
"He is a player who has a bit of everything," Dembele said.
"He has power, he is quick, he has good technical skills, and he can score goals so I am very pleased that he is now playing for us.
"He is a player who can bring a little bit extra."
Had it not been for an excellent performance from Julian Speroni in the Palace goal, Tottenham would have won by a bigger margin.
The Argentinian denied Gylfi Sigurdsson and Soldado in the first half with two top saves and he looked comfortable making the step back up to Premier League football.
The 34-year-old was impressed with how his team went about their business against Spurs and he now hopes they build on the performance when they visit Stoke on Saturday.
"We can take a lot from the performance. I don't think there was a lot of difference between the two teams," Speroni said.
"We stopped them and they are a top team, so there are plenty of positives to take.
"It's not an easy league, is it? We have to be 100 per cent."