Danny Ings capitalised on some poor Hull defending in the 62nd minute to score his 10th goal of the season, but it was not enough to prevent Burnley dropping out of the English top-flight.
Burnley manager Dyche has won many admirers for his work over the past two season, despite spending only an estimated £11 million (Dh62.4m) on players since he took charge in October 2012.
Burnley won promotion from the Championship last season and kept themselves alive in the Premier League for 36 games of a 38-match campaign.
But former Watford manager Dyche, 43, is on a rolling contract at Turf Moor and was guarded when asked about his future.
“We’re still thinking of this moment,” he said. “Next season is for other thoughts after the next few games because there are many things to be pleased about.
“Make no mistake I want to be in the Premier League so there is disappointment and frustration as well but there is a balance because when I took over this club they were 16th in the Championship.
“There’s a desire for me to get through today and see my family and make sense of everything.
“That’s another day’s story. For the moment, my thoughts are about my players.
“There will be a time to think about the bigger picture. I’m reflecting on my immediate gut feeling and that is you want to work and progress and learn.
“Wherever it takes you, who knows, but for the moment I’m Burnley manager and that’s all I’m thinking about.”
Hull’s future in the top flight is also unclear after this damaging defeat left the club in serious danger of returning to the Championship after two seasons in the Premier League.
Steve Bruce’s team were left to rue a lack of fortune and finishing as they slipped into the relegation zone for the first time since January 31.
Robbie Brady twice hit the bar with curling free kicks either side of half-time, while Ahmed Elmohamady and Paul McShane sent headers over the bar from Brady crosses.
After victories over Crystal Palace and Liverpool gave Hull hope of survival, defeats by Arsenal and Burnley have left them back in trouble.
Hull’s final two fixtures are away at Tottenham Hotspur and at home to Manchester United.
“We’ve got to go and produce two huge big performances to make sure we make a fist of staying in this division,” Bruce said.
“We have to try and get it back. We’ve still got a chance. Nobody gave us a chance against Palace and against Liverpool and we found a bit of form.”
Hull enjoyed the better of the first half but, apart from Brady’s second free-kick, rarely looked like getting back into the match as Burnley held on for their sixth win of the season.
“No disrespect to Burnley because they are a fantastic group of players who give their all, but we’ve lost to them twice,” Bruce said.
“If you’re going to stay in this league, against the relegated teams, for Hull, you have to be taking something from them and we haven’t taken anything, which has been a disappointment.”
Follow us on Twitter @NatSportUAE