Defeat leaves Simon Davey's men with work to do to ensure their survival, while Hull are on course for a first ever season in the Premier League.
In truth, Brown's men controlled from start to finish with 3,500 travelling fans backing them.
Hull could have had a flying start after just 30 seconds when they were almost gifted a bizarre opener when Ashbee clipped a clever ball forward aimed for Caleb Folan. Barnsley defender Lewin Nyatanga intercepted and headed back towards keeper Luke Steele, but the Wales centre-back watched in horror as it dropped past Steele towards the open goal before the quick-thinking Stephen Foster raced in to clear off the line.
Hull striker Fraizer Campbell flashed a warning when he looked to add to his 14 goal-tally, but the 20-year-old, who is on loan from Manchester United, was just off target.
It was another searing burst of pace that led to City's opener. He burst into the area and Foster slid in and Campbell went down. Referee Stuart Attwell immediately pointed to the spot despite Barnsley's protests and Marney sent Steele the wrong way from the penalty spot for his sixth goal of the season.
After the break, the Tykes switched from the 3-5-2 line-up that served them so well in their two impressive away wins at Watford and Preston.
But Hull were in no mood to let the lead slip and six minutes after the restart only a goalline clearance from Michael Coulson denied Campbell.
A minute later Barnsley's resistance cracked again as Hull struck with a textbook set-piece. Marney swung in a perfect corner and Ashbee charged in on cue to power an emphatic header past Steele.
Hull were denied a penalty after 61 minutes when Folan sprinted 35 yards before being felled by Dennis Souza, but amazingly the referee waved away Hull's appeals.
The evergreen Windass came on eight minutes from the end and struck with his first touch after ramming home a low shot from 15 yards.
Istvan Ferenczi headed one back in the dying seconds for the home side but it was too little, too late.