The 28-year-old striker admits to having feared for his future when it was announced that he would be among those made redundant by administration-hit Dunfermline.
He has, however, not had to wait long before bouncing back from that blow.
Barrowman has signed a deal with Dundee that will initially run until the end of the season, but he hopes he will remain at Dens Park past that point.
He said in the Daily Record: "It's a huge relief as I thought there was no chance I'd get another club for this season because of the transfer deadline, but then I got special dispensation from FIFA.
"Dundee phoned on Friday night and I came up on Sunday to sign.
"I had interest from elsewhere, including down in England, but I have decided to come here because I think it's the best move.
"I've signed until the end of the season and it gives the club a chance to have a look at me. I'll just do my best and then see if there's a chance of extending it."