Burnley had previously been the better side and created some big moments, but they were unable to beat Andre Onana in the United goal.
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Assessing his side’s display, Kompany said: “I’m like a manager that’s just lost, I wanted more but I also see a good performance so I have to balance it.
“We had two or three massive chances, really big chances. There’s the one from Jay Rodriguez, literally controlling the ball in front of the goalkeeper and slicing it, we had the one from (Jacob) Bruun Larsen one-on-one with the keeper where he just takes his touch too far, they are big moments.
“Obviously there is the one from Zeki Amdouni which just hits the post and rolls out, so I don’t think you get many more big chances against a team like United but today we did.
“We have to take the positives from it and learn from the things we can do better.”
Erik ten Hag’s side were limited to very little in front of goal as Burnley defended much better than they did against Manchester City, Aston Villa and Tottenham, where they shipped 11 goals in just three games.
But a moment of quality from Fernandes, volleying home a first-time effort from Johnny Evans’ clever first-time ball, proved to be their undoing.
“It was a pass in behind and he finishes it first time. We work on a lot on these type of moments, so we’ll have to work some more on this. We were unfortunate with the timing, definitely,” Kompany added.
“We had good pressure throughout the game, we were never exposed. If you look at our first two or three games, I don’t think this one had that type of vibe, because we were solid when we had to defend and we were good when we had the ball.
“But that moment is the quality of a Premier League player. I could say ‘this is what I would have done’, but they have to learn.”