Barrett, who started the game at fullback, replaced No.10 Otere Black with about 20 minutes remaining and put over the drop goal four minutes later to give his side a 16-12 lead.
He then added a 69th-minute penalty and winger Mark Telea finished off the best movement of the match three minutes after that to cap the 11-point burst that gave the Blues enough room to clinch victory.
The Blues had not won in Hamilton since 2011 and have now won their last six games, including a 30-20 victory over the Hurricanes in their first game of the new competition last week.
Chiefs - Penalties: D. McKenzie (4).
Blues - Tries: Hoskins Sotutu and Mark Telea; Conversion: Otere Black; Penalties: Black (2) and Beauden Barrett; Drop goal: Barrett.