The Blues Angels' biggest achievement since they were founded 42 years ago remains the Wafu Club Cup which they won in 2010.
Sharks have come close to winning the Federation Cup (formerly known as the FA Cup) on three occasions when they finished as losing finalists in 1979, 2003 and 2009.
In 1996, Sharks were also close to winning a first league title but got pipped on goal difference by Udoji United after finishing on same number of points.
But Osawe, who put in a massive shift in defence for Sharks on Wednesday when they won 2-1 at Kaduna United, believes this is the Port Harcourt club's season to end their chase for a first league title.
"Yes we can win the league (this season) because we have shown we are a winning side. What I just believe is that this season is Sharks' season.
"But we must keep our heads cool and remain disciplined because it is still a long season. With God's help and barring any injury to our players, we have a good chance of winning the league," said the former Sunshine Stars' man to Naijaligue.blogspot.com.
Osawe has also said Sharks are not under pressure to win a first league title considering the success of their city rivals, Dolphins.
Dolphins, formerly known as Eagle Cement FC and founded in 1988, have won three league titles in 1997, 2004 and 2011 as well as four Cup titles in 2001, 2004, 2006 and 2007.
The centre half is, however, unfazed by the feat of Dolphins, who have now adopted the tag Pride of Rivers, but wants to make history with the Blue Angels.
"We are not under pressure to win this league. The good thing is we have a good team to compete and one that can win the league. Sharks is a special club and all the players want to make history here but we know we have to play well and take it one game at a time," he said.
Sharks are currently top of the Glo Premier League with 14 points, same as second-placed Akwa United, but have a superior goal difference.
The Blue Angels will play hosts to El-Kanemi Warriors in a top-of-the-table clash this Sunday.