Inter, almost certain to secure their fourth straight Serie A title, were less sure back in March and also had the Champions League on their minds when they lost 3-0 at Sampdoria in the first leg.
Juve were also still in Europe and the title race when beaten 2-1 in their first leg at Lazio.
Now, after Manchester United and Chelsea dented their Champions League ambitions, the chance of a double for Inter and Juve's last hopes of a trophy this season make the unloved Coppa Italia a prize worth fighting for.
"There was a time when teams tried to avoid playing in this competition at all costs but now it has been given a new lustre and more attention is being given to winning it," Juve goalkeeper Gianluigi Buffon said ahead of Lazio's visit to Turin on Wednesday.
"We want to win it because we do not want to ruin months of work in 20-odd days."
Juve drew 1-1 at home with Inter in Serie A on Saturday, meaning the champions kept their 10-point lead with just six games left this term.
The match was marred by some Juve fans racially abusing Inter goalscorer Mario Balotelli.
The 18-year-old may be rested for Thursday's home leg with Sampdoria, who have disappointed in the league this season and need a Cup triumph to qualify for next term's Europa League.
The final will be played in Rome on May 13.