Substitute Anichebe, whose earlier header had appeared to cross the line but was not given, thought he had won it three minutes from time but Demba Ba scored his second of the night even later.
Leighton Baines had put the hosts ahead in the 15th minute only for Ba to score with virtually his first touch after coming on at half-time.
Marouane Fellaini's effort was ruled out on a questionable offside decision before Anichebe's disallowed goal added to the controversy.
Harper said: "It was behind me. I genuinely though I'd saved it but apparently it was over the line.
"You're taught never to give it up and I didn't, and we went and scored at the other end so I'm pleased.
"I got a hand on it and got a bit of luck - you need a bit of that, I didn't have much luck with the two goals so what goes around comes around."