Rumford, who shared the second-round lead with Lyle, only managed a 1-under 71 in the third round. That was enough to give him second place at minus-8.
Kevin Stadler, who won the Johnnie Walker Classic on the European Tour two weeks ago, carded a 2-under 70. He is tied for third with Paul Marantz, who shot a third-round 68, at 6-under-par 210.
Conditions were difficult on Saturday as wind gusted up to 20 miles per hour.
Lyle played very well early in his third round as he collected birdies at the par-5 second and par-3 fourth. He dropped a shot at eight for the second consecutive round, but still made the turn at 1-under-par 35.
He parred his first four holes on the second nine, then rattled off back-to-back birdies at 14 and 15. Lyle bogeyed the par-3 16th, but it did not hurt him too severely. The Australian will carry a two-stroke lead as he goes for his first Nationwide Tour victory.
'Today was as good as I've played all week,' said Lyle. 'I actually played pretty solid. I scored better yesterday, but with the conditions, today was a better round. It really is tough out there.'
Not only has Lyle never won on the Nationwide Tour, he has never held the lead with one round to go.
'This is unchartered territory for me,' admitted Lyle, a 24-year-old. 'I don't know if I'll sleep at all tonight. If the wind blows like it did today then anything can happen. I'll just try and handle the conditions as well as possible.'
Scott Dunlap (66), Wade Ormsby (67) and last week's Jacob's Creek Open Championship winner Paul Sheehan (68) are knotted in fifth place at minus-5.
Ryan Armour (67), Adam Crawford (69), Dave Moreland IV (69), Paul Gow (71) and Jason Dufner (71) share eighth place at 4-under-par 212.