“When I play for my country I just think about trying to win the next point, trying to fight really hard for my team-mates, for everybody who is here and everyone who is watching from home,” said Berrettini.
“That’s the biggest privilege. If I win it’s better, but I try to enjoy being out there every time. It is something so special.”
World number 22 Cobolli followed his compatriot with a commanding 6-1 6-3 victory over Austrian number one Filip Misolic.
The 23-year-old Cobolli, who impressed with a run to the Wimbledon quarter-finals this year, dominated a first set lasting just 30 minutes with three breaks of serve.
He tightened his grip on the match early in the second set, securing an early break as he moved into a 3-0 lead, before confidently serving his way to victory.
Cobolli echoed his team-mate’s sentiments in his post-match interview, saying: “We all want to win this cup and I felt the support of the bench, it was unbelievable.
“When I play for them, it is something which I cannot describe.”