With dominance in the forward pack, at the set piece, and with relentless defence, the visitors' contained the Drua's pace and flair.
Overcoming a slow start, the Auckland side brushed aside a number of handling errors exacerbated by the hot and humid conditions to lead 20-7 at halftime.
First-five Barrett kicked a penalty to open the scoring after 25 minutes when neither side posted any points on the scoreboard.
Then Rieko Ioane opened the try-scoring for the Blues, before the Drua, supported by a vocal home crowd, hit back through No.8 Elia Canakaivata.
The Blues' big man Cameron Suafoa scored on the strike of halftime, taking him to three tries for the season.
Defending continually to open the second half, the Blues captain Dalton Papalii grounded the ball for the visitors' only try of the second half as the Drua kept the contest alive.
With a great start from the line-out, Drua raised the stakes as Canakaivata crossed in the corner earning his double to close the gap to 27-14.
Barrett guided the Aucklanders back to a position of some strength as he slotted a penalty from out in front.
And the Rebels survived a tenacious comeback from a fast-finishing Moana Pasifika, 43-33, in a bruising encounter at Mt Smart Stadium.