Sinclair has this week been in talks with the Red Devils and he has now signed a two-year contract with Richie Barker's side, who are Posh's opponents at London Road on Saturday.
He could make his debut against Posh in that game.
Crawley are thought to have beaten off competition from Notts County, Swindon and Shrewsbury to sign Sinclair, who previously turned down a move to Rotherham.
Sinclair has scored 13 goals in 49 appearances, 21 as a substitute, since joining Peterborough from Macclesfield in August 2011 on a three-year deal.
He scored three Championship goals for Darren Ferguson's team last season, all of them coming in a 3-1 victory at Hull, and spent time on loan at Doncaster and Barnsley.