The former Swords skipper held his nerve to score a fantastic 32 not out that earned the three-times finalists a one-wicket, last-over win over Bridgwater before a bumper North Parade crowd.
Bath will host Ealing in the last eight on Sunday, August 8 having beaten the Middlesex Premier League outfit by five wickets at the same stage of the competition a year ago.
And their success completed a weekend double over the Somerset side after the Globe Sports West of England Premier One leaders extended their advantage at the summit with a thumping 178-run success in Bridgwater.
Backed by a big away following for the cup clash, Bridgwater were restricted to 146 all out by an excellent fielding effort from Bath after electing to bat first in the last-16 cup clash a day later.
Six players claimed wickets, with captain Tom Stayt (2-29), Joe Dorgan (2-30) and Lloyd Davies (2-22) taking two apiece. Adam Kelly (1-22) bowled well and ran out Chris Skidmore with a direct hit from the boundary.
Bath found the going tough in reply, however, after losing Robin Lett for a duck early on.
His fellow opener, James Campbell, top-scored with 34 and Alex Barrow made 17 but, after Ross Jenkins was run out for two, Bath slumped from 75-4 to 90-7, with Stayt exiting for a first-ball duck.
Adam Kelly and Brown swung the game back in Bath's favour with a key stand of 37 for the eighth wicket but when the former was controversially given out to a catch from Stuart Butt, Bath were left needing 20 to win off 24 balls.
Joe Dorgan joined Brown and helped carry Bath to the brink of victory before he was run out following a mix-up between the pair from the last ball of the penultimate over.
Bath required two to win with Brown at the striker's end and, amid mounting tension, he hit the fourth ball of the last over over mid-off for four to seal a memorable triumph.
Saturday's league meeting was a somewhat more one-sided contest as an unbeaten century from Ben Staunton (102no) helped the unbeaten table-toppers post 250-7 before they shot the home side out for just 72.
Stayt scored 43 and then took 3-24 in Bridgwater's reply, before spinner Vikram Banerjee finished the hosts off with 3-10.
Weston-super-Mare are the visitors to North Parade on Saturday.