The 27-year-old forward is the latest addition to Brendan Rodgers' squad and had previously worked with him at Swansea.
Sinclair, who could go into the Hoops squad for the Ladbrokes Premiership opener against Hearts at Tynecastle, told the club's official website: "It feels amazing to have joined Celtic
It's a massive club and I'm looking forward to it
I'm here now and I'm happy to be here."
The fee paid to Villa is reportedly around £3million, which could potentially rise to £4.5million with add-ons.
Sinclair is hoping to make a quick impact at Parkhead.
He said: "I can't wait to get on the pitch
When I went back to Aston Villa for pre-season, I heard Celtic were interested, and I just knew I had to come here.
"I had to look at this move and I thought it would be great for me.
"And because I know the manager and I know what to expect, hopefully I can get off to a flying start."
Indeed, Sinclair claims he played his best football under Rodgers after signing for Swansea in 2010.
He then moved to Manchester City in 2012, and then Villa three years later, having already enjoyed a loan spell in the Midlands.
"The manager had a great influence when I was at Swansea and that's when I played my best football," said the former England Under-21 international.
"And hopefully now I can get the best football out of me and get my form back out there and score goals and perform.
"I'm just hoping to come here, score some goals, create chances and hopefully excite the fans.
"I know Kolo Toure and Dedryck Boyata from Manchester City, but I'm sure it's a great set of lads here and I'll quickly settle in."
Sinclair watched Celtic's dramatic 2-1 midweek Champions League win over Astana and claimed the prospect of European nights at Parkhead was one of the reasons he decided to join the Hoops.
He said: "I watched the Astana game in midweek and the atmosphere was incredible.
"To have the possibility of Champions League football excites me and I can't wait to come here and play at Celtic Park and for it to be rocking."
Source : PA
Source: PA