Brown kept his head when Leeds had been allowed back into the game after two clearance kicks - one apiece from Matt Powell and Giscard Pieters - had been charged down, resulting in second-half tries for Andre Snyman and Tim Stimpson.
Having squandered a 13-point lead, Worcester had their backs to the wall and Leeds were back to within a point. But from 16-15, Brown banged over two vital penalties to settle Worcester's nerves and determine the outcome.
The significance and manner of Worcester's win was not lost on their head coach, Andy Keast, who said: "We know we can compete with the best and the score didn't reflect our superiority. The whole squad has a never-say-die attitude."
From the opening exchanges, at a packed and vocal Sixways, the Leeds scrum was in dire difficulties. Having missed with his first penalty, Brown, when offered another kickable chance, opted for a line-out drive. The Worcester forwards rumbled towards the posts for Tony Windo to get the touchdown. Brown converted and the Warriors were on their way. Although Stimpson pulled three points back with a penalty, Brown struck twice with drop goals and added another penalty before the half was finished.
But the defining moment of the half came when Leeds flanker Andrew Parks was sin-binned for foul play. Mercifully for his team, they only shipped three points while Parks was in the cooler.
Leeds clawed their way back into the game after the break, but Mark Regan was then yellow-carded and their revival all but disappeared. The home forwards celebrated by re-establishing their momentum and the Tykes were gone.
Worcester: Try Windo Conversion Brown Penalties Brown 3 Drop goals Brown 2. Leeds: Tries Snyman, Stimpson Conversion Stimpson Penalty Stimpson.
Worcester: T Delport; D O'Leary, D Rasmussen, T Lombard, G Pieters (G Trueman 63); J Brown, M Powell; T Windo, A van Niekerk (C Hall 69), C Horsman, P Murphy (T Collier 68), C Gillies, D Hickey, P Sanderson (capt), B Macleod-Henderson.
Leeds: I Balshaw; A Snyman, P Christophers, C Bell (D Rees 55), T Stimpson; G Wright, ADickens; M Shelley (K Fullman 45), M Regan, R Herber, C Murphy (S Hooper 55), T Palmer (capt), S Morgan, R Parks, C Rigney (P Uys 65).
Referee: D Pearson (Northumberland).