Chasing the Hurricanes' 6-170 for their first win of the BBL season, the Sixers looked long odds until a stunning final over from Billings got them within a hair's breadth.
Needing 18 from quick Tymal Mills' final three balls, Billings hit consecutive sixes onto the hill, leaving a maximum for victory off the last delivery.
But all he could produce was an air swing from a wide Mills slower ball.
Billings was unbeaten on 61 from 31 as the Sixers finished agonisingly short on 4-165.
The Hurricanes third win in a row means they leapfrog Sydney Thunder into fifth place on the ladder, while the Sixers are now 0-6.
Earlier, opener D'Arcy Short top scored for the Hurricanes with 42 from 36.
The tournament's leading run scorer smacked five boundaries including a straight six that reached the second tier of the grandstand.
Short shared a 77-run stand with wicketkeeper Matthew Wade (41 from 30) for the second-wicket.
Wade and Short departed in the 11th and 12th overs respectively, as the Hurricanes' innings spluttered.
But some late hitting from veteran allrounder Dan Christian (28 from 14) and Simon Milenko (22 from 10) pushed the total to 8.5 runs an over.
The pair took a particular liking to seamer Daniel Sams, who finished as the Sixers' most-expensive bowler with 1-49 from four overs.
Sydney's chase got off to a sluggish start until Billings combined with Jordan Silk for a 60-run partnership.
Silk was caught in the deep off Mills in the 17th over, who returned to close out the game, albeit just.