Luke Chambers had given the home side the lead early in the second half, but a late Rose header earned his club a share of the spoils at Portman Road.
Ipswich responded with a chance for Chambers, but he put his effort over the bar from six yards out.
Barnsley should have taken the lead just moments before half time when Marlon Harewood hit the post
Despite the visitors' momentum, it was the hosts who took the lead 11 minutes into the second half, when Chambers turned on the edge of the box before firing in the opener.
However, with just moments left to play, second-half substitute Rose popped up to snatch a late equaliser for Barnsley with a header.