Icardi equalised with a sweet 85th-minute volley before Matias Vecino completed the hosts' quickfire turnaround to secure a much-needed 2-1 victory at San Siro.
Inter's skipper had been largely anonymous until his equaliser and finished the match with just 16 touches, the fewest of any starter.
"[It takes] a bit of everything," he said.
"He must make movements that will benefit the team, just as they must consider what his qualities are, putting him in a position to express himself better.
"A few seconds of talent are enough to be a top footballer, but to be a champion it takes 95 minutes of personality."
Inter's success against Spurs came despite an unconvincing start in the league which has yielded just four points from as many matches.
Spalletti hopes the confidence gained from the Champions League will lead to a consistent run of results.
"If we're able to win these games, we come out of them stronger because it means that we're capable but we have to keep doing it," he said.
"We also benefitted from the enthusiasm after such a win. It could be that we follow up on the win tomorrow and to do so would mean much more than winning two individual matches.
"I hope that the guys become aware of their abilities."