The Clarets netted inside the first minute at Turf Moor.
The Whites could, and perhaps should, have picked up at least a point at Turf Moor on Saturday as striker Chris Wood missed two fine chances, sending a couple of second-half headers wide of Tom Heaton's post.
In the end, Scott Arfield's first-minute winner settled a tight affair, but Cooper is drawing solace from the idea that United outplayed a side which look destined for Premier League promotion at the moment.
Having said that, Steve Evans's side remain winless from four Championship outings now, and the Scotland defender was blunt in his assessment that it is just "not good enough".
“We need to do better,” said Cooper, as quoted by the Yorkshire Evening Post. “We need to take our chances and keep chances out. We can’t start like we did. The boys are hurting and I thought we played very well. We’re disappointed not to come away with anything but we gave ourselves a big task conceding so early. Coming away to the league leaders, you can’t do that.
“The positive is we’ve been the better team. We did enough to get something out of the game and maybe win it but overall it’s disappointing. The first minute of the game, away from home, against the league leaders – you can’t do it. It’s a sin. On a whole, as a team we’ve got to do better. That’s no win in four and it’s not good enough.”
United have now lost three of their last four games - a 1-1 draw with QPR being the exception - as Evans's men sit 15th in the Championship table.
Eleven points separate them from relegation but the Scottish boss, who replaced Uwe Rosler in October, could perhaps be doing more to strengthen his claim for a new contract, with his existing deal set to expire in the summer.