Simon Dyson, who took the early lead Saturday, posted a 4-under 67 and stands alone in second place at minus-14.
Eight-time Order of Merit winner Colin Montgomerie carded a 2-under 69 and shares third place with Soren Hansen, who also shot 69 Saturday. They stand at 9-under-par 204 with Jean-Francois Lucquin (69) and Emanuele Canonica (70) two strokes back at minus-7.
Brier fell one shot behind Dyson as the Englishman birdied the first. Dyson though bogeyed the fifth, then Brier took the lead with an 18-foot birdie putt on No. 6 at Fontana Golf Club.
The Austrian hit a remarkable approach to the par-5 ninth as his second shot rolled within 5 feet of the hole. Brier drained the eagle try to jump to 13 under, three clear of Dyson.
Brier parred his next three holes, then holed a 108-yard, sand wedge approach shot for eagle on the par-4 13th.
'I said to my caddie, I want to hit this past the pin because it's always going to come back,' Brier stated. 'I know the course pretty well and knew the pin was tucked on the left. I didn't want to go left of the pin.'
Brier two-putted for birdie on the par-5 15th, but gave that stroke back with a three-putt bogey at the next.
'Today with the strong winds, I missed a few greens but made some really good up and downs fortunately and that kept my round going all the way,' Brier said. 'I birdied the ninth, then that holed wedge was lucky, but it was a nice shot. So that's how my score came together.'
He closed with a pair of pars. Brier, who won this event twice when it was part of the Challenge Tour, will be looking for his first win on the European Tour on Sunday.
After falling one behind Brier with his bogey on No. 5, Dyson fell further back as his approach at the eighth missed the green right. He was unable to get up and down for par and the bogey dropped him to minus-9.
Dyson, who played alongside Brier, quickly got that stroke back as he got up and down for birdie on the ninth, his third birdie in three rounds on that par-5. He closed within one of Brier with back-to-back birdies from the 11th.
The Englishman pitched his third at the 15th to 3 feet and dropped that putt in for birdie to move to 13 under. Dyson two-putted for birdie at the par-5 closing hole to finish within one of Brier's lead.
'The winds were strong out there so you had to strike the ball good. Out of the three days, that's the best I've hit the ball definitely,' admitted Dyson. 'I thought I hit good putts on 16 and 17, but neither fell. We were both helping each other and hopefully we can continue that on Sunday.'
David Higgins (72) and Gregory Bourdy (70) are tied for seventh place at 6-under-par 207.
Miguel Angel Jimenez struggled to a 3-over 74 Saturday to slide into a share for ninth at minus-5. He stands alongside Gary Emerson, Richard Green, David Griffiths and Alex Rocha.