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 appeared to connect with the ball before Campbell went down.
Referee Stuart Attwell immediately pointed to the spot despite Barnsley's protests and Marney crashed the ball past Steele 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.
Zoltan Gera fired West Bromwich Albion back to the top of the Championship with a priceless strike against arch-rivals Wolves in a frenetic encounter at Molineux.
Gera's 59th minute winner has put Albion within touching distance of a place in the Premier League, but dealt a serious blow to Wolves' hopes of reaching the play-offs.
It also continued a dismal run for Wolves against Albion as Mick McCarthy's side have now lost five of the seven league, FA Cup and play-off meetings between the sides in the last two seasons.
Albion were well worth their success as they twice hit the post through Gera and Kevin Phillips, while Wolves found chances hard to find.
Once they fell behind their best opportunity to salvage a point came from midfielder Seyi Olofinjana, but his 88th minute drive was well held at his near post by Dean Kiely.