The Bergamaschi's campaign began with a 2-2 draw at Genoa while Palermo claimed an impressive 4-3 triumph against Inter Milan.
"Palermo will be extremely motivated after beating Inter," Cigarini said. "We will have to be a million times more motivated than them if we are to be successful.
"We did well in Genoa but we only got a draw and hence, in order to win we need to do better."
There is a sense of urgency for Atalanta to start winning as soon as possible.
The Bergamo outfit gained promotion to Serie A last season but a summer betting probe saw Stefano Colantuono's team start the new campaign with a six-point deduction.
"We have to get rid of that penalty immediately," Colantuono said. "It's not easy to start a season like this.
"But we are a competitive squad and I'm convinced that we can do well."
Palermo, meanwhile, hope to pick up where they left off against Inter when they travel to the Stadio Atleti Azzurri D'Italia.
The Sicilian giants came twice from behind to stun last season's runners-up last weekend.
The result handed their new coach Devis Mangia a perfect start.
Mangia replaced sacked Stefano Pioli last month.
"I could not have asked for a better result," Mangia said. "But if the game against Inter was difficult, our next game will be even more so.
"It's not easy to win in Bergamo and we will have to show the same determination we displayed against Inter."
The visitors will be without injured midfielders Luca Di Matteo and Adam Simon.