Southend, who wanted to keep him, had said that a bid "had been tabled at a level that all sides in League Two would not be able to match".
The former Cambridge United and Histon player made just eight starts for Posh since turning his loan switch from Histon into a permanent move in 2009.
He has been on loan at the Shrimpers since the beginning of the season, scoring twice in 24 appearances, but was scheduled to return to London Road on 4 January.
It was a big decision for him but he really does believe he has made the right choice and I certainly do
Crawley boss Steve Evans
Red Devils boss Steve Evans told the club website: "Many times in the last week I thought we had lost out on getting Josh Simpson here but we kept working hard and hoping we could land him.
"I have met with him on several occasions, every minute of my time and every mile driven was worth it when he telephoned me to say he would love to join us and what we are trying to achieve here.
"The lad was fortunate in that he had several clubs wanting him and it was a big decision for him but he really does believe he has made the right choice and I certainly do.
"He is an intelligent lad and loves playing football. The reality is that he could have stayed at Peterborough United and earned a lot more money there but the kid wants to play first team football and feel valued, he will get both here."
Source: BBC Sport
Source: BBC Sport