Although the transfer fee paid for Jennings is undisclosed, the non-league side had made it clear they were looking for a figure of more than £100,000 for the player.
Jennings, 20, has scored 50 goals in 78 starts in his two years of senior football - including scoring on his debut against Hinckley.
"I feel I'm ready for the step-up to League One. I've just got to train hard, get to know my bearings and then take it from there," he said, speaking to the Iron's official website.
"There is a week until the next game and that will be a big help for me now. I'll get to know the players first and get used to the training, and it's something I'm really looking forward to.
"I want to carry on the form I've been showing with Stalybridge, so if I can do that I'll be very happy. This season has been my best one yet and hopefully I can carry on that form here."
Jennings has signed until the end of the 2014 season, and has been handed the squad unmber 28.
He becomes the first signing of the January transfer window, as manager Alan Knill looks to refresh his side to try and head off a relegation battle.
The move came ahead of Chris Dagnall completing a move to the Championship side Barnsley, where he will again work with manager Keith Hill - as he did at Rochdale.
The Tykes have paid an undisclosed transfer fee for Dagnall.
Defender Michael Nelson is also expected to leave Glanford Park, with his agent claiming the Scottish Premier League among possible destinations, while the club also hope to bring Shane Duffy back on loan later in the month despite his parent club Everton recalling him to face Tamworth in the FA Cup third round.