Nabil Bentaleb put Tottenham in front in the 18th minute of the League Cup quarter-final at White Hart Lane - his first senior goal for the club.
Nacer Chadli netted at the beginning of the second half to double the advantage, before Harry Kane and Roberto Soldado added further gloss.
Newcastle rarely looked like challenging for the match as Tottenham stormed into the semi-finals, and midfielder Stambouli says the goals came naturally.
"I think it was a perfect performance," he said. "We were focused on our opponents, we had that willingness to score and it came naturally.
"It is never easy even if you are winning 2-0, but we focused on our defending and then our attackers took their chances.
"All the club and all the fans are very happy. We will do our best to win this trophy."
Tottenham will now face League One outfit Sheffield United in the last four, and head coach Mauricio Pochettino has vowed not to take Nigel Clough's men lightly.
"We have respect for Sheffield United - it is never easy," he added. "Yesterday they played very well and deserve to be in semi-final.
"We expect a tough game."