The Argentine was sacked after a dismal start to the season and replaced by Jose Mourinho.
Speaking after Spurs' third straight win since Mourinho took charge, the Belgian defender admitted Pochettino was axed because of their poor performances.
He told reporters: "Our displays [under Pochettino] were not up to our standards.
"We're all responsible. He put us on the pitch to win and we haven't been doing that like we should, so everyone feels responsible."
Vertonghen also claimed to have passed on his regards to Pochettino, adding: "I sent him a [private] message. That was my message to him."