Paul Anderson had opened the scoring in the 20th minute with a crisp shot into the corner but when Cristian Roberts fired in an angled drive in the 70th minute to equalise, it was just reward for the Robins.
Feeney rose above a static defence at the far post five minutes later to nod in a cross from Tom Butler to gain all three points for the Swans.
The Swans were gifted the opening goal when a poor pass out of defence was gathered by Jason Scotland and he whipped in a cross from the left that evaded Robins goalkeeper Phil Smith, leaving Anderson to race in and fire home.
Darryl Duffy suffered a recurrence of his groin strain and was replaced by Feeney for the start of the second half but when Roberts levelled for the Robins it looked like more home points lost by the Swans only for Feeney's header to send the crows into euphoria.
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