Three minutes later, the Hammers made it 2-0 with an outstanding finish from Lanzini, who flicked the ball to his left foot and volleyed home after good work from Declan Rice.
Lanzini made it three just before the break, drilling home a penalty after a VAR review found that Luka Milivojevic had handled in the box.
Palace got a goal back in the 83rd minute when Odsonne Edouard stretched to turn in a fine cross from substitute Michael Olise who then scored himself in the 90th minute with a free kick.
Patrick Vieira's side had hit the woodwork twice in the first half and enjoyed the bulk of possession and Antonio was generous in his praise of Palace.
"To be fair this season Crystal Palace are playing unbelievable football. The way their gaffer has them playing is quality," Antonio said.
"All game they were after us, but when opportunities dropped for us we took them. They managed to get two and we had to hold out. You saw the pressure they put us under but we managed to hold out and it's three points to start the year."
The meeting between the two sides in August had ended 2-2 and Palace defender Joachim Andersen said it was hard result to take.
"We've played West Ham two times now and both times we have dominated them. I think we should win this game easy. They had two chances and a penalty which isn't a penalty. We should win this game," he said.