A second-half strike from Ashley Barnes was enough to give the Premier League's bottom side their first win of the season on Saturday.
Bruce's men looked noticeably off the pace throughout a disappointing outing at Turf Moor and mustered just one attempt on target during the 90 minutes.
The manager's misery was compounded as Curtis Davies was forced off with a back injury after Bruce had made three substitutions, leaving Hull to chase a win with 10 men.
"It's very rare in the two-and-a-half years I've been at the club that I can be embarrassed and say sorry to supporters who have travelled on an awful afternoon in their thousands," Bruce said.
"I can obviously look at myself in making three substitutions before 60 minutes, which was obviously a gamble, but I have to say we were probably better with 10 men than we were with 11.
"That's a sad indictment of the first hour in particular where we just nowhere near the level which has seen us stay up in the Premier League and have a decent start to the season.
"That performance has eroded all we've done because it was simply not good enough.
"We knew we didn't want to be the first one [to lose to Burnley], but to be fair we've gifted Burnley their first win because we didn't kick a ball in the first half."
