The Blues scored the only try of the first half, Hoskins Sotutu burrowing over after his fellow forwards laid siege to the Chiefs line.
That score was largely against the run of the opening 40 minutes, with Damian McKenzie's unerring boot keeping the Chiefs in touch as their opponents' penalty count mounted.
Goalkicker Otere Black made way for the Blues after the hour, allowing Barrett to move forward to five-eighth.
The switch paid instant dividends as the All Blacks superstar landed a 64th-minute drop goal. Five minutes later, his penalty opened up a seven-point advantage.
The Blues did not look back from that point and Mark Telea finished brilliantly in the corner to make the game safe.
Barrett was unable to add the extras from the touchline but his influence had already proved decisive in securing a franchise record fifth consecutive win away from home.