Carlisle had an early chance when Danny Graham turned neatly in the box inside the first minute, but found Darren Randolph racing out to block the shot. Long-range efforts from Graham, Danny Carlton and Graham Kavanagh then kept the home defence on their toes.
At the other end Hereford battled away but their chances were few and far between, with Clint Easton lifting a shot over from 30 yards and Williams having a low drive saved by keeper Ben Alnwick.
Carlisle looked the more likely side to break through and Carlton got past Randolph only to see his angled shot booted away from danger by skipper Karl Broadhurst. The former Bournemouth defender came to Hereford's rescue again when he did well to block a shot from Carlton.
Hereford started the second half strongly and former France international Bruno N'Gotty surged forward to test Alnwick from 30 yards. The on-loan Leicester defender was then instrumental in forcing the corner on the hour which brought the decisive goal. Easton's flag kick was half cleared to Williams, who lashed an unstoppable shot into the net from 15 yards.
Williams forced a low save from Alnwick after good work from Sam Gwynne and the Bulls continued to look dangerous, but after Keogh was dismissed, Carlisle, despite their numerical inferiority pressed forward strongly. Substitute Jennison Myrie-Williams fired wide and then into the side-netting but the Hereford defence, in which N'Gotty and Broadhurst were outstanding, held firm.