Gary Hooper got things going on 15 minutes with a simple finish after finding himself unmarked in the penalty area.
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Victor Wanyama scored a brace in six minutes to put the match beyond doubt before the break, following up a neat finish with a curling effort from distance.
St Mirren regained a measure of composure in the second half but could not find a way back into the match.
Tony Watt added his name to the scoresheet five minutes from time, heading home a pinpoint cross from Emilio Izaguirre.
Wanyama came close to completing a hat-trick in injury time when his powerful header crashed against the woodwork.