The last of those wins, a 2-0 victory at Sampdoria on Wednesday, gave the club its best-ever start to a league season.
The 100 per cent start has fuelled optimism among fans that Roma may have what it takes to capture the title, something they have failed to do since 2001 when Fabio Capello was head coach.
But Giallorossi sporting director Walter Sabatini and his French coach Rudi Garcia are stressing that it is not how a team begins, but how it finishes.
"The Scudetto is not something that we are competing for," Sabatini said.
Nevertheless, he concedes that everything has worked to perfection so far, with the team scoring 12 goals and conceding only one.
"We are playing well," he added."We want to do well and to make Roma fans happy.
"We are fortunate to have found a coach that has empathy with the team and the fans."
Garcia, who arrived from Lille in the summer, added: "It is good to have won five games but what really interests me is entering the history books at the end of the season, not now."
Roma captain Francesco Totti is expected to start against Bologna on Sunday, having played as a second-half substitute at Samp, but the Giallorossi will be without injured full-back Maicon.
And Garcia, who was ordered from the touchline early in the second half at the Marassi, will have to watch Sunday's game from the stands as he serves a one-match ban.
Bologna coach Stefano Pioli, meanwhile, is still seething over the late collapse that allowed AC Milan to claim a point on Wednesday.
The Rossoblu were searching for their first win and led 3-1 with five minutes remaining, but goals from Robinho and Ignazio Abate spoiled the party at the Stadio Renato Dall'Ara.
"There is still a lot of bitterness about the result," Pioli said. "I was very satisfied with my team's performance until the 85th minute.
"It is very disappointing to have been so close to the win only to be denied in the end.
"But we must leave this setback behind us and focus on our next game, which will be very difficult."