Monza had taken the lead in a summer friendly against Everton, but were eventually held 2-2 with goals from Andrea Colpani and Dany Mota Carvalho.
The club made their first ever Serie A appearance last season and finished mid-table, so are aiming even higher this time around.
It was a strong start to the friendly match played behind closed doors at Goodison Park, as Colpani’s curler thumped the upright, with Patrick Ciurria testing the goalkeeper.
Colpani did eventually get his goal on a splendid free kick at the dawn of the second half.
Matteo Pessina and Maupay tested the goalkeepers before Everton turned it around with a well-worked Lewis Dobbin goal and Neal Maupay sent clear by a defensive error.
Monza snatched the 2-2 draw again from a set play, this time Jose Machin’s free kick finding the head of Dany Mota Carvalho.
Everton 2-2 Monza
Colpani 52 (M), Dobbin 70 (E), Maupay 76 (E), Mota Carvalho 78 (M)
Everton: Virginia; Higgins (Welch 30), Godfrey, Branthwaite, Mykolenko; Garner, Onana; Onyango, Dobbin, Danjuma (Kouyate 77); Calvert Lewin (Maupay 46)
Monza: Di Gregorio; Izzo (Cittadini 62), Caldirola (Bettella 88), A Carboni (F Carboni 73); Ciurria (Birindelli 73), Gagliardini (Bondo 88), Pessina (Machin 62), Carlos Augusto (Kyriakopoulos 73); Colpani (V Carboni 73), Caprari (Mota 73); Maric (Petagna 62)