McInnes' side took their Premier League opposition the distance over two legs, but eventually it was the Englishmen who prevailed 4-2 on aggregate.
"Obviously I would like to congratulate Burnley first and foremost," McInnes said. "It was a fantastic game, the intensity over the two matches has been remarkable. You don't get those types of games at this stage of the season," he said.
"With the rivalry of the Scotland-England thing, both sets of players have clearly been very competitive with intensity from both sides and both sets of supporters. The Burnley supporters and our supporters helped generate what played out tonight.
"My first thoughts are disappointment for my players when they put so much into it but there is no criticism.
"We cannot afford to be too disappointed. We now move on to the domestic game and Rangers on Sunday.
"I thought it was harsh for us to come out, somebody had to come out as losers and somebody had to win, it's cup football and Burnley got the all-important goal in extra time."