Hakan Çalhanoğlu, Yann Bisseck and a second-half brace involving Francesco Pio Esposito did the damage at San Siro, though Chivu admitted his side had needed to shake off first-half rustiness before finding their rhythm.
"In the first half we lost a bit of sharpness, making a lot of technical mistakes," Chivu told Inter TV. "In the second half we found our composure again, and the team responded well both technically and in terms of the way we played. We're working to improve, that's the starting point.
"We're happy with the new arrivals, we hope they settle in quickly. We have to keep working at this level, trying to improve, because there's still important room for improvement."
Asked specifically about the performance of Inter Milan's summer signing Amadou Diouf, Chivu felt there was more to come from the winger despite his role in the opening goal.
"A bit shy today, for me he can do better. We struggled to find the right combinations, but he has the qualities to make things happen in one-on-one situations. On Pio's goal he showed real initiative. There's still important room for improvement there too."
Monza, beaten in their first match back in Serie A, took some encouragement from a first-half showing that briefly troubled Inter before the hosts pulled away.
Coach Ivan Jurić said he had been pleasantly surprised by how his young side had coped.
"I'm very happy, I didn't expect we could play with so many youngsters like that. In the first half we deserved more, I'm really pleased."
Asked why he had left out Patrick Cutrone, Jurić kept his answer brief: "I chose to set up the tactics this way."
Jurić pointed to the six absentees his side were missing as context for the display.
"We have to think that today we were without six players. We need to be complete across all the positions, I remain satisfied with what I saw today. We'll try to keep playing this way and take games on with an open approach. We went 2-1 down without Inter really creating chances."
The Monza coach was pleased with a surprise selection at centre-back, handing a start to young defender Lucchesi.
"He did well, we could have made a change but I liked what I saw. I tried him out, he's a lad to build with. He can improve."
On further transfer business before the window closes, Jurić did not hide his hopes.
"Today we were missing three midfielders. I expect a lot from the market, a huge amount. If we want to stay up we need to strengthen in several areas."
He also praised a debut showing from Mout, a player he admitted knowing little about beforehand.
"I didn't even know him myself, he's made impressive strides. In a short time he's understood what's needed. Like a lot of young players he has big room to improve."
Asked about goalkeeper Pizzignacco, beaten four times, Jurić was sympathetic.
"Yes, I feel for Pizzignacco. He was unlucky, there are small details that decide everything in Serie A."
On left wing-back Mangas, deployed further forward than usual, he added: "I put him up front out of necessity, he's naturally a left-sided wing-back. He's about transitions and intensity. We expect a lot from him."