Summer signing Roberto Soldado was on target at Selhurst Park but Jermain Defoe and Gylfi Sigurdsson were culpable of missing the target when one-on-one with home goalkeeper Julian Speroni.
Spurs were almost punished late on when they had goalkeeper Hugo Lloris to thank for keeping them in the game with a couple of fine saves and Vertonghen insists they must improve.
"We were very dominant, but we kept them alive and they had a big opportunity in the 89th minute," said Vertonghen.
"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."
Recent recruits Paulinho, Nacer Chadli and second-half substitute Etienne Capoue also did well and Mousa Dembele feels Tottenham have a good chance of improving on their fifth-place finish in the Premier League last season.
"We had quite a good season last year but at the end we didn't make what we aimed for," he 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."