Mazzarri's side went into Sunday's encounter sitting ninth in Serie A after an underwhelming start to the season, with the match coming at the end of a turbulent week.
Thursday saw honorary president Massimo Moratti resign - the 69-year-old reportedly unhappy at the club's continuing support of under-fire Mazzarri.
Inter were also held to a draw at home to Saint-Etienne in the UEFA Europa League, and Mazzarri revealed that his players were beginning to feel the heat.
After seeing Mauro Icardi's 32nd-minute penalty edge out 10-man Cesena, who had goalkeeper Nicola Leali controversially sent off for his role in conceding the decisive spot-kick, Mazzarri told Sky Sport Italia: "We really needed this win, as the lads felt the pressure.
"I would remove the rule of the red card for the goalkeeper, as I like a game with 11 against 11 and we had the same sort of thing happen to us in Turin last season.
"You could tell the team was not relaxed and felt the pressure. We were too hasty to score and therefore didn't make the most of the chances, often wandering offside because we didn't hold the run.
"I know they were desperate to win and prove themselves, but at times you have to be calmer and take your time to make the right decision.
"We could even have shot from distance, but seemed determined to walk it into the net."
Leali's dismissal for a coming-together with Rodrigo Palacio - who seemed to make the most of the contact - ensured Inter had a numerical advantage for almost an hour, but Mazzarri did not feel his side took full advantage of the extra man.
He added: "In fact, we played better before the red card, as we became frenetic and wasted several chances which were one-on-one.
"We cannot keep wasting these opportunities and I got angry with a lot of things, as we wasted so many counter-attacks.
"Anyone who [has] played football knows that at 1-0 you can always concede at any moment, even from a ricochet, so you have to finish the game off.
"We could've finished it without running those risks in the final minutes. Cesena kept pumping high balls into the box and we really risked blowing it."
