Marcello Lippi's side could only draw 1-1 with Paraguay in Cape Town on Monday as they came from behind to salvage a point.
They are now under pressure to get all three in their next game against New Zealand on Sunday but Fiorentina playmaker Montolivo believes they can improve after a tough start.
"The first match is important but we were too stressed about the start of the World Cup," he admitted.
"I'm happy with the match, I had a good shot which the goalkeeper saved in the second half, we had a lot of concentration but were stressed about the start of the World Cup.
"Coach Lippi told us we have to try shots on goal but we were not too lucky in this match. We conceded a goal against Paraguay but once it was 1-0 to them we came back a bit to defend and make sure we didn't go 2-0 down. We didn't want to lose the first game.
"The effort and heart we put into this game, I think it shows we can win the next games."
Defender Domenico Criscito, deployed at left-back in an otherwise familiar back four against Paraguay, admits it will take some work to get back to the resolute performances of the 2006 FIFA World Cup-winning side.
"We were not very happy with the result against Paraguay because the team can do better," he insisted. "I was very happy to make my World Cup debut, it's great. We had some problems with the defence and the next match with New Zealand will be very important."