Sam Vokes and Danny Ings were both denied by Kelvin Davis, who came on after 10 minutes to replace the injured Fraser Forster.
Dyche was also dismayed by referee Roger East's decision not to award a penalty when George Boyd appeared to be fouled by Jose Fonte.
Shane Long made Burnley pay in the 37th minute with a clinical finish and a Jason Shackell own goal after the break wrapped up the points for Southampton on manager Ronald Koeman's 52nd birthday.
"We came down here, played on the front foot and created four good chances and should have had a penalty," said Dyche.
"That's three we haven't been given in recent weeks now and I have no problem with referees, but that was a penalty."
Dyche praised replacement goalkeeper Davis, but knows his side missed a chance to boost their survival prospects.
"I played with Kelvin many moons ago and he is a fine keeper, but it was more our poor finishing," added the Burnley boss.
"They were a good height for the keeper and we have to learn that at this level, you have to take those chances.
"I was a bit disappointed with that. I rarely question my strikers because I want them to play with a freedom to miss chances, but I felt they were very good chances that we should have taken.
"The second half was flat until they got the goal and then it’s tough for us, but we kept trying to effect the game and that's how we play."
