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Rhodes pounced on a goalkeeping error to open the scoring for the Tartan Army after 10 minutes.
The Blackburn Rovers striker doubled his tally 10 minutes later, reacting quickest to score a rebound after Steven Naismith's drive was saved.
Gilks was forced to make a fine save from Stefano Bensi with 20 minutes remaining, turning a strong header past the post.
Rhodes thought that he had completed his hat-trick late on but his third goal was ruled out for offside.
Bensi tested Gilks again from range in the closing stages but the Scots hung on to secure an unconvincing victory.