Cosmi's men endured a dismal start to the Serie A campaign but two wins and a draw in their last three games have fuelled hopes the Salentini can climb out of the drop zone.
They are third-bottom but go into Wednesday's clash at the Stadio Friuli looking to pick up where they left off in Sunday's 1-0 triumph over Inter Milan.
"We are hungry for wins," Cosmi said. "We never stopped believing that we could turn things around.
"Last weekend we faced a difficult game against Inter but we showed a lot of maturity against a team that were also determined to get a positive result.
"My players gave their all to take the win."
Lecce have shown an improvement both in attack and defence recently, with four goals scored and two conceded in their last three league games.
"We are more lucid, more focused now than last year," Cosmi said.
"My team is now able to strike at the right time.
"Of course victories help the team gain confidence and work in harmony during the week."
Cosmi knows there is still a lot of work to do, however.
"I think our recent run of form is a step forward,'' he said. "But we cannot dream.
"We must be realistic and not stop. The path to avoid relegation is still a long one and we know we will have to pedal hard.
"In recent weeks, we have sent strong messages to our opponents but we have a lot more messages to deliver."
Udinese will be looking to bounce back from Saturday's 2-1 loss at Juventus.
The result was Udinese's second defeat in three games and left the Friuli side third, six points behind league leaders Juve.
Udinese remain unbeaten at home in the league this season, with Francesco Guidolin's men having collected 28 points of a possible 30 at the Stadio Friuli.
The hosts will be without suspended defender Pablo Armero and injured midfielder Giampiero Pinzi.