Trump opened up with a break of 106 to open up a one-frame lead but McLeod regrouped to secure the next frame with runs of 70 and 15.
McLeod had his hand on the table first in the next but could only make a 42, which allowed Judd in and runs of 8 and 67 were enough for him to secure the frame.
McLeod took a scrappy fourth frame 57-40 but then world number six Judd claimed the next three frames – with another century break coming in frame six – to move into the next round.
Joining Judd in there will be Shaun Murphy, who overcame Jack Lisowski 5-0.
Murphy took the first frame 62-45 before breaks of 90 and 103 took him to a three-frame advantage and the 2005 world champion easily claimed the last two frames to progress.
Australian Open results
Martin Gould 5 - 3 Michael Holt
Neil Robertson 5 - 2 Andrew Higginson
Peter Ebdon 5 - 2 Barry Hawkins
Judd Trump 5 - 2 Rory McLeod
Alan McManus 5 - 4 Joe Perry
Shaun Murphy 5 - 0 Jack Lisowski