In a match in which yellow was very much the colour of the day, French referee Cedric Marchat sent eight players to the sin bin - six of them from the Italian club.
Flankers Maxime Mbanda and Nicola Belardo, centre pairing Tommy Castello and Andy Vilk, tighthead prop Lorenzo Romano and scrum-half Simon Piconealso all saw yellow for Calvisano while Newcastle lost Will Welch and Joel Hodgson in the second half.
Newcastle wing Zach Kibirige got their first try after 24 minutes and full-back Noah Cato their second two minutes later.
That suggested a deluge of tries for the home side but instead they had to wait until the 50th minute to bank the bonus point, number eight Chris York getting try number four after the Falcons had earlier been awarded a penalty try.
Left wing Michele Visentin and number eight Braam Steyn did cross for the visitors but Cato collected his second and the Falcons' fifth in the 66th minute.