The England squad member was all square with Australian Brett Coletta through 13 holes of their second-round match.
But Bower, twice a winner in South Africa this year while touring with England Golf, won the next three holes with the aid of two birdies and a par to progress.
Masham's Dan Bedale and Jonathan Thomson, of Rotherham, were Yorkshire's other players who made it through to the match play stage from two qualifying rounds of stroke play.
Thomson was beaten 5&4 by Switzerland's Jeremy Freiburghaus in the first round and Brown fell to another Swiss, Marco Iten, 2&1 in the second round.
It meant Yorkshire were denied an all-White Rose last-32 clash, as Bower faces Iten on Thursday morning.
Richmond's Karen Jobling lies second, three shots behind the leader Sue Spencer (Whittington Heath), after two rounds of the English senior women's stroke play championship at Burnham Beeches.
Huddersfield's Pat Wrightson (82 84) and Hallamshire's Val Swales (81 89) also made it through to Thursday's final round.
Another Hallamshire player, Pauline Bramley, missed the cut by two shots (83 89).