The champions were not at their best until after Anthony Stokes capitalised on a mistake by Saints defender Steven Anderson in the 55th to fire the visitors ahead.
To the anger of the Saints players and fans, skipper Dave Mackay was shown a straight red card in the 75th minute by referee John Beaton for a foul on Hoops substitute Derk Boerrigter inside the box with Nir Biton converting from the spot-kick.
The award was all the more galling for the home side as Mackay had claimed for a penalty minutes earlier after a challenge by Celtic defender Virgil van Dijk in the box.
Hoops midfielder Callum McGregor added a third in the 85th minute with a low drive which keeper Alan Mannus should have saved to compound a miserable night for the Scottish Cup winners.
The Hoops boss, who was "pleased" with Craig Gordon's competitive debut in goal, said: "The first 20 minutes was okay and the last 25 of the first half was not the way I want to see my team play.
"But in the second half we upped the pressure and intensity and it opened up and we got the chances and won the game.
"The second half is more like I want to see my team but we need to work on the fitness and pattern. But that will come.
"I am not unbelievably happy because I didn't think we performed fantastic but we are on our way.
"It was not perfect but the second half was a step in the right direction."
The Norwegian was also happy with James Forrest's first start of the season but urged the winger to look to the work-rate of superstars such as Arjen Robben and Franck Ribery.
Deila said: "He is unbelievably important but he is only 50 per cent of what he can do.
"He had 20 minutes working defence and offence and then he stopped working defence and if you are to get his potential out, he can look at Robben or Ribery and how much they run.
"But he has played 60 minutes and 15 or 20 minutes against Legia so he needs time. He is a very important player for us."
St Johnstone boss Tommy Wright claimed referee Beaton got both referee claims wrong and will consider appealing Mackay's red card.
He said: "One was a penalty and one wasn't. Simple.
"It's not my view that counts, it is the referee's that counts.
"There is enough grounds for appeal and I have spoken to John.
"Van Dijk leaves his leg out. There was contact. It is the wrong decision.
"If there was contact (with Boerrigter) it was minimal and it didn't make him go down because he didn't go down in the contact, that is clear for everyone to see.
"And he actually collapses his right leg which is the leg furthest away from Dave Mackay, if he did make contact.
"I have to concentrate on performance and that was excellent."
Deila said of the incident: "I haven't seen it but I will say the same all season. If it is against us I hope I will be calm and say after 38 matches it will work out even.
"The referees do their best, sometimes they make mistakes, sometimes they will do good things like the players and I will do."