Neil Lennon's side were held to a 1-1 draw inside 90 minutes but were finally able to find their way into the last eight of the competition thanks to James Forrest's winner in extra time.
Fans weren't happy with the performance of the 'treble treble' winners, but Boyd defended them, saying that they were undone by poor officiating from referee John Beaton.
The 53-year-old claimed Celtic are "always" hampered by the man in the middle.
He said: "We've not played well. But when you're playing 12 men it's tough. And we always are."
Celtic have won the last nine domestic honours in Scottish football and have won eight consecutive league titles.