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The victory was the icing on the cake for the Ibrox club as they were presented with the Scottish Third Division trophy after the match.
Aird's diving header from a Kyle Hutton cross gave Rangers a first-half lead, although Scott Dalziel threatened an equaliser.
Lewis MacLoed and Darren Cole came close to extending the Gers' advantage, but in the end the solitary goal was enough for the victory.
Berwick won't be too downhearted as they finish the season in fourth place to progress to the promotion playoffs.