The world No.1 hardly broke sweat against the 22-year-old wildcard, who shocked Beatriz Haddad Maia in Round 2, easing into the last 16 in 53 minutes.
Barty claimed a 15th successive victory and will meet Alison Riske in round four, after the American knocked out No.13 seed Belinda Bencic.
Australian Barty wasted little time in the first set, and she was 5-0 up within 20 minutes, having already displayed an impressive array of shots, with a backhand down the line to secure the fifth game a particular standout.
Dart managed to hold her serve to avoid a whitewash, though Barty swiftly regained control in the next game to seal the set.
The French Open champion broke Dart twice as she kept up the pace in the second set and took the first four games.
After holding serve, the Briton looked poised to get one of those back, but failed to take any of four break points in the sixth game, with Barty converting her first match point in the next.
Ashleigh Barty  bt Harriet Dart 6-1 6-1
Barty - 23/6
Dart - 5/18
Barty - 11/2
Dart - 0/5
BREAK POINTS WON
Barty - 5/5
Dart - 0/4
FIRST SERVE PERCENTAGE
Barty - 64
Dart - 57
PERCENTAGE OF POINTS WON ON FIRST/SECOND SERVE
Barty - 92/50
Dart - 48/32
Barty - 56
Dart - 27