The 21-year-old, who featured in the final 16 matches of the league season, was expected to leave the club on a free transfer next month.
However, the Dublin-born shot-stopper has now signed a new two-year contract, with the option of a further year based on appearances.
"I think Ian knew in his heart he was going to be here next year and I wasn't worried for one minute. I was confident he would come to Ireland with us on the pre-season tour and that he would want to further his career with us," Robinson told the club's official website.
"Ian is definitely one for the future so I am excited to have acquired his services for another two years. He is someone who I think will go from strength to strength as a goalkeeper and he is a wonderful character to have around the place.
"He did extremely well last year, after settling into the first few games he settled into our style of play, he worked with me and Paul Heald to improve on his feet and his passing and he became a real asset to us, which is why we were keen to keep him at the club beyond the summer."
A number of Football League clubs had previously been linked with his services.