The newly-promoted Magpies claimed their second win of the campaign by beating the Swans 1-0 at the Liberty Stadium.
Sunday's result, secured thanks to a second-half Lascelles header, edged Newcastle into the top 10 of the table, and the defender was delighted with the team performance.
Lascelles told reporters: "The boys were excellent today. We knew it would be a tough game. Swansea like to play football but we knew we could get at them and had a game plan.
"I have got great boys around me and great support. We are very close and it follows through on to the pitch.
"These are the teams we need to beat. They are a good side but we were the better team on the day. the Premier League is tough and that is why we need to be on it every week."
The result was also Newcastle's first away win in the Premier League since December 2015.