Barrett converted all three of his own tries and kicked three penalties as the All Blacks utility back led his team to their first win of the season.
The 24-year-old, playing at fullback, put his side's first points on the board after six minutes with a penalty before scoring the opening try three minutes later.
Barrett finished off a straightforward move that saw the Hurricanes win their own line-out and move the ball through the backs before crossing the Highlanders' line.
Despite trailing, the Highlanders dominated possession throughout the opening exchanges but were unable to put points on the board.
They were given the perfect opportunity to haul themselves into the game when the Hurricanes' prop Tyrel Lomax was sinbinned for collapsing a maul short of the try line, but the home side struggled to press home the advantage.
Barrett showed the defensive side of his game on the half-hour mark when his superb last-ditch tackle denied Freedom Vahaakolo inches from the line to keep the Highlanders scoreless.
Tony Brown's team did notch their first score when Bryn Evans crossed the line with 30 seconds remaining of Lomax's sanction and, with Josh Ioane converting, the deficit was reduced to three points.
Barrett, though, was on hand again to extend the Hurricanes' lead when he finished off a period of sustained pressure after an offload from Salesi Rayasi to score his second try.
Barrett struck again four minutes into the second half, bursting through the defence to score his third try before kicking a penalty from inside his own half to put the Hurricanes further ahead.
Tries from Connor Garden-Bachop and Thomas Umaga-Jensen kept the Highlanders' slim hopes alive, but the visitors capped the scoring with Barrett's late penalty.