The second-tier champions are five points shy of safety following their 2-1 home defeat to Sampdoria and the run-in includes tricky games against the likes of Fiorentina, AC Milan and Juventus.
Di Francesco said: "We knew this was a crucial game for us and we played well, putting some good moves together and always trying to keep the ball.
"Unfortunately we weren't clinical enough in front of goal. The boys have to hold their heads up high, we have to give everything we've got."
Paolo Cannavaro, Matias Alexe, Francesco Acerbi, Matteo Brighi, Luca Antei, Thomas Manfredini, Nicola Sansone and Antonio Floro Flores will all miss Sunday's meeting with Roma through injury.
Roma this week unveiled plans to build a new 52,500-seater stadium - the 'Stadio della Roma' - that the capital club hope will be ready for the 2016-17 season.
The president of the second-placed Giallorossi, James Perotta, is plotting Roma's "re-emergence as a force in international football" and sees the new stadium as being integral to realising that dream.
On the pitch, Rudi Garcia's men retained a slim chance of catching leaders Juventus, who are 14 points better off, after beating Torino 2-1 on Tuesday night.
But French coach Garcia is keeping one eye on the exploits of third-placed Napoli and said: "We'll have to wait a bit before we secure second place mathematically.
"What I can say is that we still have a big lead over Napoli."
Full-back Federico Balzaretti is unlikely to feature while the season is over for Dutch midfielder Kevin Strootman, who suffered serious knee damage earlier this month.