Free agent Lita has signed a one-year deal after being released by Swansea City and should make an impact at this level.
Barnsley certainly needed the former Boro forward and Owls loanee in the first half when Dale Jennings was the only Reds player to threaten, cutting inside from the right and forcing a diving save from Brian Jensen.
Crawley had twice gone close in that tepid goalless first period, Izale McLeod flashing a header wide and captain Josh Simpson striking the base of a post.
A last-ditch challenge by Joe Dudgeon meant Simpson was unable to continue in the second period.
Barnsley’s hopes were hit when Jennings received a straight red in the 62nd minute for grabbing Gwion Edwards by the throat in retaliation and the Reds misery was complete in the 81st when an unmarked McLeod headed home the winner from five yards from a corner.