The winger scored the winner on his debut for the club on Saturday.
Sinclair, signed from Aston Villa in a deal worth a reported £3.5 million on Saturday, came off the bench to score a late winner for the Hoops in their opening game of the Scottish Premiership season against Hearts on Sunday.
He was engulfed by the travelling supporters as he celebrated with his teammates, and looks to have immediately endeared himself to the Parkhead faithful.
The 27-year-old admitted that he wants to create a special bond between himself and his new fans, and said that he is loving his first experience of Scottish football after being persuaded to reunite with Brendan Rodgers.
Sinclair said, as quoted on Celtic's official Twitter account: "The manager had a big impact in me coming here. He knows me and has the confidence to play me freely.
"Celtic fans are some of the most passionate in football. I want to enjoy my football again and bring back the excitement of playing."
"I think [Scottish football] is great. I want to get a bond with the fans, make an impact and help us win. As a new player it's always nice to get off the mark. It's a great start for me and now I want to build on that in each game."
Sinclair will be keen to finally kickstart his career again after enduring a frustrating four years since leaving Swansea City for Manchester City in 2012.
At Villa last term he managed six goals in 27 appearances in all competitions but was never truly a regular for the relegated club.
He will be hoping that he finds a new lease of life in Glasgow, and he has certainly made the best possible start.