The cup holders' 2-0 defeat in Naples on Wednesday meant they failed to reach the semi-finals for the first time in nine years.
The setback also snapped Inter's eight-match winning streak in all competitions, but Chivu believes it can spur the Nerazzurri on in the league.
"It's a real shame that we are no longer alive in the Coppa," the former Romania international said to www.inter.it.
"But I'm confident this will give us more determination to do well in Serie A and in the Champions League.
"We believe we are in the race to win the Scudetto.
"We are six points behind league leaders Juventus and there are still 19 games left to play so anything can happen."
Claudio Ranieri's men beat Lazio 2-1 last weekend to climb to fourth place.
The Nerazzurri are still without midfielder Dejan Stankovic and striker Diego Forlan, both injured.
Moreover, midfielder Thiago Motta, who is a reported Paris St Germain, is doubtful with a thigh problem.
The mood in the Lecce camp is sombre ahead of tomorrow's match, with the Salentini having failed to win at home this season.
Last weekend's 2-2 draw against Chievo left Lecce in 19th place, six points adrift of safety.
"It's possible to avoid the drop but we must improve our performances at home - perhaps starting already on Sunday against Inter Milan," goalkeeper Massimiliano Benassi said.
Midfielder Manuel Blasi's return to the club this week from Parma is a consolation for Lecce coach Serse Cosmi.
Blasi, who previously played for Lecce in the 1998-99 season, is fired up to give his new side an immediate contribution.
"I want to give my best to help Lecce climb up in the standings," he said.