The Scottish champions were more than holding their own until Dutch defender Virgil van Dijk was sent off in the 36th minute after picking up the second of two yellow cards, the first for a foul on Rodrigo Palacio, the next for tangling with Mauro Icardi.
It was an uphill task for Deila’s side thereafter and although they battled hard, Fredy Guarin fired in a wonderful strike with two minutes remaining to cap a frustrating evening for the Scottish champions.
The Celtic boss described the sending off as “very cheap” and said of the Slovakian official: “He had a bad day at work, that’s for sure.
“If we had that bad a day, we would have lost 6-0. There were things that happened in the game that ruined it for us. It was very even when 11 against 11. We defended well with 10 men but couldn’t create anything.
“I feel sorry for the team — but very proud of the boys. It was an open match until the sending-off.”
“Inter are a good team so we have to get over it and start focusing on what is going to happen on Sunday.”
Deila was referring to the visit of second-placed Aberdeen in a match which will go some way to deciding the destination of the title.
The champions are three points clear of the Dons with a game in hand and they face Dundee United in the William Hill Scottish Cup quarter-final and Scottish League Cup final next month as the quest for the domestic treble continues.
Roberto Mancini admitted Van Dijk’s dismissal gave his side an advantage, but insisted they had the game in hand.
The Inter boss said: “It made our life easier but we had the chances to score before that. We managed the game in the right way.”
INTER MILAN: Carrizo, Santon, Ranocchia, Juan Jesus, D’Ambrosio (Campagnaro 80), Guarin, Medel, Hernanes (Kovacic 80), Palacio (Puscas 89), Shaqiri, Icardi.
CELTIC: Gordon, Matthews, Denayer, van Dijk, Izaguirre, Mackay-Steven (Ambrose 40), Brown, Biton, Armstrong (Commons 78), Johansen, Guidetti (Forrest 59).
Ref: Ivan Kruzliak (Slovakia).