Montolivo scored the only goal in Sunday's 1-0 victory over Catania with a first-half stunner at San Siro, as Milan won a fourth straight Serie A win to move up to eighth in the table.
They sit five points off rivals Inter, who occupy the final Europa League place, with one of their five remaining games being the second Milan derby of the season on May 4.
Italy international Montolivo feels his side may have to win all of their remaining fixtures to qualify for Europe, but stated belief was high in the camp under Clarence Seedorf's guidance.
"We can't afford any more slip-ups this season, we have to win every game," the skipper told the club's official website.
"It'll be tough but we can do it. I needed that goal and I'm happy it was a decisive one.
"It'll be tough against Livorno but we have to go for the win. The Europa League is still distant, we can't be making calculations and we have to think only about ourselves.
"We believed in this comeback and day by day the work we're doing is paying off."
