The three points came at the U-Power Stadium thanks to goals from Lorenzo Pellegrini and Romelu Lukaku towards the end of the first half before Paulo Dybala's stunning free-kick in the 63rd minute and a late Leandro Paredes penalty.
Andrea Carboni netted a fine consolation goal for 11th-placed Monza with a long-range rocket in the 87th minute of Roma's sixth win in seven league fixtures since De Rossi replaced Jose Mourinho in January.
Roma are fifth, one point behind Bologna who sit in the final Champions League spot and are at top-four rivals Atalanta on Sunday evening. Roma have scored 20 league goals under De Rossi.
"I'm just managing a group of human beings as much as a group of players. I try to be honest and direct when I see something that I don't like," said De Rossi to 'DAZN'. "I'm really happy to be working with them, both for how they perform and how they make me feel."
De Rossi was a surprise choice to take over at Roma after Mourinho's sudden sacking, but has so far risen to the task. The only league match Roma have not won under their former captain was against runaway leaders Inter Milan, a 4-2 defeat but one that came after a fine performance.
Saturday's win was perfect preparation for the first leg of their Europa League last 16 tie with Brighton on Thursday, which will be played in front of a packed and loud Stadio Olimpico.
Roma reached two consecutive European finals under Mourinho, who is beloved by fans for ending a 14-year trophy drought with the inaugural Europa Conference League in 2022. De Rossi will have star attackers Dybala and Lukaku in good spirits after a strong display from the pair in northern Italy.
Dybala's free-kick was his ninth goal since the turn of the year and his 12th in the league
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