Bradley Dredge and Simon Khan finished two shots back at 12-under-par 204. Gary Murphy, Jamie Spence, David Drysdale and Stuart Little followed at 11- under-par 205.
Webster was four shots behind Finch to start the day at Castello di Tolcinasco Golf Club and tallied his first birdie of the round at the par-5 first. Webster then played his approach to the fringe at the par-4 fourth, but drained the putt for another birdie.
The Englishman bogeyed the fifth, but countered with a birdie at the par-4 seventh after his second shot stopped 5 feet from the cup.
At the par-5 ninth, Webster two-putted for another birdie and kept on rolling with a birdie at the very next hole. He hit his approach to 6 feet for a birdie at the par-4 11th and played his third shot out of a bunker within a foot of the hole for a birdie at the par-5 12th to make it four in a row.
Webster was in a bunker again at the par-4 13th, but knocked his second shot inside 6 feet. He converted the putt for his fifth consecutive birdie to join Finch in the lead at 15 under.
Webster's run came to an end at the par-3 14th when he three-putted for a bogey. The 30-year-old answered at the par-5 fifth and reached the green in two en route to another birdie.
He flew his second shot over the green into a bunker at the par-4 17th and walked away with a bogey. Webster kept it together with a par on the closing hole, however, to secure the 54-hole lead.
'It's a little bit frustrating towards the end,' said Webster, who has the lead entering the final round for the first time on the European Tour. 'I made a lot birdies around the turn and I felt like I was swinging great.'
Webster, who will be seeking his first win on the European Tour, has finished second five times.
'I hope I just keep it going tomorrow,' he said. 'Tomorrow's a different day. We'll see what happens.'
Finch was in control early with three birdies over his first five holes. He gave a shot back with a bogey at the par-3 eighth, but took two putts for a birdie at the par-5 ninth to make the turn at 15 under.
The Englishman ran off five straight pars on the back side before struggling to a bogey at the par-5 15th. Finch then bogeyed the 18th for a round of 71.
Anders Hansen, Emanuele Canonica, Marcel Siem and Maarten Lafeber share ninth place at 9-under-par 205.