A composed penalty in the first half and then a spectacular drive from distance in the second helped the Clarets bounce back from last week's home defeat to Wigan.
Geovanni, who thought he had equalised for Hull with a free-kick which the referee ruled out, was then sent off to cap a miserable week for the visitors.
The game, the first ever Premier League meeting between the two sides, began at a frantic pace.
Dead ball expertise: Alexander scores from the penalty spot
The hosts did get some early joy down both flanks with Chris Eaglesand Robbie Blake providing testing crosses. But despite plenty ofattacking intent, neither side's goalkeeper was really troubled.
Hull's only real chance was a shot from Seyi Olofinjana which he did not quite catch on the edge of the box.
The opener arrived in the 19th minute when the referee gave a softpenalty and Graham Alexander marked his 100th appearance for Burnleywith a goal.
The spot kick came when Stephen Hunt clumsily brought down Tyrone Mears after a fine pass from Elliott.
The Hull players surrounded the referee but Alexander kept his cooland blasted the ball into the bottom-left corner of Duke's goal to givethe hosts the lead.
Hull, clearly lacking in confidence having taken just one point onthe road this term, pushed on in pursuit of an equaliser and somecommanding goalkeeping from Jensen helped Burnley weather the pressure.
Seeing red: Geovanni is sent off for Hull
The Clarets carved out another golden opportunity with half an hourgone. A fine pass from Andre Bikey sent Eagles scampering down theright. Blake took up a clever position at the near post but hisattempted chip sailed over Duke's crossbar.
Elliott had a thunderous drive beaten away by Duke at the near postand Eagles was then denied by a smart stop as half time approached.
It was an untidy start to the second period but Hull created the first chance and should have pulled level.
Striker Jan Vennegoor of Hesselink slotted the ball through forOlofinjana but instead of shooting the Tigers midfielder opted to playthe ball square for Kamel Ghilas.
The 25-year-old forward, who was already under pressure before he got the ball, fired straight at Jensen.
Hot under the collar: Phil Brown observes proceedings
A fine run and shot from substitute Bernard Mendy, which saw himbeat several Burnley defenders on his way, was initially fumbled buteventually gathered by Jensen.
And Hull then had a goal ruled out in the 67th minute for pushing inthe wall when Geovanni curled home a free-kick which Jensen could onlypush into the roof of his net.
And just a minute later the Tigers midfielder was given his marchingorders for a clumsy challenge on Fletcher which saw him pick up hissecond yellow.
The Clarets pegged their struggling opponents back with Eagles orchestrating much of their attacking work down the right.
Hard fought battle: Hull's Andy Dawson and Chris Eagles of Burnley collide
And a second, which was always on the cards following Geovanni'sdismissal, came in the 77th minute when Alexander fired home inspectacular fashion from 25 yards.
Substitute Kevin McDonald found the evergreen midfielder in space,he advanced completely unchallenged and unleashed a powerful drivewhich nestled in the bottom-corner of Duke's goal.
Burnley introduced Fernando Guerrero during the closing stages andhis pace proved a constant threat. But the scoreline remained the samedespite the hosts having plenty of chances to extend their lead.
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