Spurs struggled to find their rhythm for long periods at Selhurst Park, with Christian Benteke proving to be a tricky customer.
The north Londoners' back three initially failed to get to grips with the Belgian striker but Pochettino shuffled his pack ahead of the interval.
Eric Dier was moved forward into midfield while Mousa Dembele and Victor Wanyama were both withdrawn at the break as Spurs effectively matched up their opponents for the second half.
The tactical tweak proved to be a success as Christian Eriksen's effort handed Spurs a crucial victory and Vertonghen is adamant they were good value for all three points.
"We both played at the weekend," Vertonghen told Tottenham's official website.
"[In the] second half you could see they were a bit tired - they played on Sunday and maybe they're not used to it.
"We played better and they never crossed our line in the second half.
"The first half was a bit more difficult. We changed system as well and that was the right solution."
Spurs are now just four points adrift of pacesetters Chelsea, who cruised past Southampton on Tuesday night.
Palace have beaten several top sides in recent weeks and Vertonghen was delighted he and his teammates answered the questions asked of them by their title rivals.
"Chelsea put pressure on us with their victory and everyone thought 'go to Palace who have beaten all the big teams and show us what you've got'.
"We've progressed a lot since last year and we had a very professional win," the Belgian added.