In what was their 11th Premier League game of the campaign, the decisive moment came for the Clarets in the 50th minute when Barnes connected with Kieran Trippier's cross.
They might have scored earlier, with Danny Ings shooting narrowly wide early on before being denied by a Steve Harper parry, the loose ball then being sent back towards his own goal by Paul McShane and cleared off the line by Robbie Brady.
They were then playing against 10 men for the final 20 minutes - Hull captain Curtis Davies having been forced off with an injury when his side had already made three substitutions - but Sean Dyche's team could not add to their tally.
That will not bother Burnley fans, who had not celebrated a Premier League win for their team since May 2010.
And while Dyche's side remain bottom of the table, they are now only two points from safety, having taken their total to seven.
Hull - who have not scored in three straight games - stay two points clear of the drop zone in 14th place.
We use your sign-up to provide content in ways you've consented to and to improve our understanding of you. This may include adverts from us and 3rd parties based on our understanding. You can unsubscribe at any time. Read our Privacy Policy