Dredge managed a 3-under 68 Saturday at Crans-sur-Sierre, while Rumford posted his second consecutive, 5-under 66. The pair finished at 13-under-par 200.
Overnight leader Oliver Wilson only shot a 1-under 70 in the third round and is one shot back in pursuit of his first European Tour victory.
Sion E. Bebb fired an 8-under 63 on Saturday and is tied for fourth place with Spain's Gonzalo Fernandez-Castano, who shot a 6-under 65. The duo is knotted at 11-under-par 202.
The final threesome of Dredge, Rumford and Wilson seemed like the walking wounded on Saturday. Wilson was battling a left shoulder injury that caused him to seek treatment before the round and Rumford complained of a headache.
Dredge was healthy and proved it early on Saturday. He eagled the par-5 first hole for the second consecutive round, but dropped a shot at the par-4 fifth.
He rolled in a 10-foot birdie putt at the sixth and added another at seven to go to 13 under, two clear of the field. Dredge missed the green with his second at the par-5 ninth, then chunked his third into the bunker. He could not save par from the trap and fell to minus-12.
Dredge got the stroke back with a 6-foot birdie putt at the 10th, but made a mess of the 12th. His fourth shot at the par-4 ran nine feet past the hole and his bogey effort stayed above ground. He made double bogey and fell into a huge tie for the lead.
Dredge and Rumford distanced themselves at the par fives on the back nine. Dredge lagged his 50-foot eagle try two feet short and tapped in for birdie at 14. Rumford holed a slightly longer putt for birdie to match Dredge in the lead at minus-12.
Both Dredge and Rumford found bunkers at the par-5 15th. Dredge nearly holed his pitch from the sand, but settled for a tap-in birdie. Rumford blasted out to 8 feet, but sank his birdie putt to keep pace with Dredge.
Dredge converted a 6-footer for par at the 16th and both parred out to share the 54-hole lead.
Dredge will be in search of his third victory on the European Tour. He has the chance to become the first player to successfully defend his title at this event since Seve Ballesteros in 1977-78.
Rumford will also look for win No. 3 on the European Tour. He tied for second earlier in the season at the New Zealand Open and has made five cuts in a row, but is 106th on the Order of Merit.
Wilson was 1 over on the front nine, but back-to-back birdies at 14 and 15 got him within one.
Lee Westwood (65) and Phillip Archer (68) are tied for sixth place at 10-under 203. Anthony Wall carded his third straight round in the 60s with a 3-under 68. He is alone in eighth at minus-8.
Richard Finch (64) and Miles Tunnicliff (70) share ninth at 7-under-par 206.