Leicester's Harry Maguire earned the quickest Premier League red card in four years when he was dismissed in the fourth minute for bringing down Johann Berg Gudmundsson, but the visitors still took the lead in the 33rd through a free kick from James Maddison.
Teenager Dwight McNeil equalized with his second league goal five minutes later and Burnley pushed for what would have been a potentially crucial win in its battle to stay up. But instead it was Morgan who made the breakthrough, heading in Youri Tielemans' deflected cross to hand Burnley a fourth straight loss.
Burnley remained two points ahead of 18th-place Cardiff, which has a game in hand. Leicester is 10th.
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