Sean Dyche felt the “edge” was missing from his side as a Tom Heaton own goal proved enough to earn Swansea all three points at Turf Moor.
The Clarets remain in the bottom three of the Premier League after failing to make the most of a tight game that looked destined for a stalemate.
And Dyche was left to rue coming out on the wrong side of the Premier League’s “fine margins”.
He said: “It’s a frustrating one because the general play, apart from being just off with that real edge we play with, was good.
“I was certainly pleased with the way we were trying to win the game because that was apparent.
“It was just the fine margins of the Premier League I felt, but we know we have to turn performance into wins.
“There is no naivety to my positivity; we know we have to win games because points are hard to come by in this league.”
Heaton barely had a save to make all afternoon other than the goal, when he first denied former Claret Jack Cork, then kept out Kieran Trippier’s skewed clearance, only to see the ball spin over the line as he attempted a third save.
At the other end former Arsenal stopper Fabianski made crucial saves in either half from Ashley Barnes and substitute Sam Vokes, who was the late subject of a penalty appeal.
Dyche added: “Neither side were on top of their game. There was a bit of a flatness to both sides.
“In the second half we came out really well and I don’t think they had been in our box for 20 minutes and they get a corner and it’s a soft goal.
“If you don’t make first contact you have to make the second and we didn’t make either. Tom makes a great save, ‘Tripps’ reacts and it unfortunately spins off his knee.
“Tom makes another great save and unfortunately it just squirms out of his hands.
“It was a soft one and a frustrating one because we were on the front foot and creating half chances, not necessarily golden chances.
“To get the goal go against us was a blow, but we kept going and almost got something.”
Commenting on the two penalty appeals, when City defender Federico Fernandez inadvertently handles, then Vokes looked to be impeded by Neil Taylor, Dyche insisted: “I don’t think their lad can do a lot about the penalty shout when it hits his hand.
“The second one on Sam I have certainly given. We pride ourselves on our honesty, and at times maybe too honest because Sam just jumps and gets pulled down enough to stop him going for the ball, but doesn’t make a meal of it.
“I don’t want him to because I want us to play with that honesty, but I think we are all aware of watching other games how many players would make a big meal of that and then the referee is under pressure.
“You can only ask the referee to make decisions and I thought that certainly had a better than even chance of being a penalty.”