The hosts were soon level as Jay Rodriguez finished from close range after receiving a cross from Ian Maatsen.
Story of the match
Vincent Kompany made three changes to a Burnley side that were defeated by fellow relegated side Watford last time out. Jack Cork, Vitinho and Rodriguez were named to the starting XI with Dara Costelloe, Manuel Benson and Ashley Barnes on the bench.
Rodriguez was out of the traps early, a half-volley attempt from 20-yards away going just wide of the right upright.
Their aggressive style nearly cost the Clarets as a short back pass from Charlie Taylor saw Estupinan clear, but it hit Arijaenet Muric and went wide.
At the other end, Tobias Figueiredo slipped, allowing Rodriguez' shot from an acute angle, Ingram did well to block with his chest.
On 25 minutes, Hull were in front. A volleyed clearance by Figueiredo went to Estupinan, who glanced a header to Tufan who had made an unmarked run through the centre and Muric.
Maatsen saw his volley held by Ingram, but Burnley's next passage of play led to the equalizer as Josh Cullen's sublime pass to Maatsen was crossed by the Dutchman to Rodriguez, who tapped home from six yards.
In the closing moments of the first half, the lively Maatsen nearly registered his second assist of the evening, floating a cross that Cullen met, but headed wide.
Three minutes into the second half, Tufan played an inch-perfect pass to Estupinan, who was only denied from a goal-line clearance by Taylor Harwood-Bellis.
The Burnley defender then made a surging run, playing the ball to Vitinho through Connor Roberts, but Ingram held on to the Brazilian's effort.
The hosts nearly went in front in the 64th minute, a Rodriguez flick finding Brownhill, but his strike could only hit the side netting as the Clarets continued to waste good opportunities.
Brownhill then brought down a lovely lofted pass by Cullen and he set up Rodriguez, his shot at Ingram lacking power and conviction, untroubling the 'keeper.
Muric then had to be alert to a fierce drive by Benjamin Tetteh and at the other end, Maatsen teed up Benson, but he went woefully wide to sum up the hosts' troubles in front of goal.
Player of the match: Taylor Harwood-Bellis
While Burnley had 77 percent of the possession and had relatively little to contend with in their own end, when they did, the on-loan Manchester City centre-back was always in the perfect position and didn't give the ball away. A top-class performance.