The match was marred by controversial moments, including a 10th-minute flashpoint which saw Celtic's Virgil van Dijk receive a red card while referee Craig Thomson sent off United's Paul Paton instead of Calum Butcher in a case of mistaken identity.
And the first half was coming to a close when Connolly went to ground in the box after minimal contact from Celtic's Anthony Stokes.
Nadir Ciftci converted the spot-kick, but after the game Celtic boss Deila hit out at 19-year-old Connolly.
"It was a dive, in my opinion
I think we are the only team in Scotland who has had a suspension because of diving," Deila said.
"We had Derk Boerrigter suspended earlier in the season for diving (he was banned for two games after diving to win a penalty in a 3-0 win at St Johnstone in August).
"It's no problem if you're tackling every incident that's happening, but I haven't seen one more suspension for diving, so hopefully it applies to all teams.
"This was a tough game to be a referee
Now it's just about doing things right afterwards."
Celtic were awarded a penalty of their own in the 48th minute when Paul Dixon blocked a shot with his arm in the box and was sent off, but Radoslaw Cierzniak came to the rescue with a superb stop to deny Leigh Griffiths' penalty.
The United 'keeper made a series of superb saves before Griffiths headed home the equaliser in the 71st minute.
Deila felt his side had enough pressure in the second half to avoid a replay, though.
"In the second half we had a lot of chances and should have killed the game," the Celtic manager said.
"But it was a tough game and we are still in it and the next game is at home, which is a positive thing
There are going to be some exciting games coming up."
- Mistaken identity -
After an uneventful start to the match, events took an extraordinary turn in the 10th minute with the bizarre sendings off of van Dijk and Paton
There was a flashpoint in midfield involving Celtic's Scott Brown, who slid into a challenge on United's Ciftci before Butcher and van Dijk clashed
However, after consultation with his assistant referee, Thomson flashed the red card at the innocent Paton as well as van Dijk.
"I think what was strange to me was that the initial challenge from Brown didn't get dealt with by the referee
I think if that gets dealt with then the rest doesn't happen," a perplexed Dundee United boss Jackie McNamara said.
"Once it broke out I didn't see what happened in the melee but for me the initial challenge on Ciftci was a bad one.
"I feel sorry for Paton
He got suspended earlier in the season for something he didn't do and he's been picked out there again
He must think everyone is against him.
"We will absolutely appeal the red card and we'll see what happens, but I think common sense will show Paton wasn't even involved in it.
"The players are also saying that Dixon's red card was harsh and it has maybe just caught him on the elbow, but the referee has made a decision and we have to accept it.
"It's disappointing because I felt at 10 versus 10 we had a real opportunity to win the match but this was a turning point.
"But I'm proud of the players in the way they responded on going down to nine men and the crowd played their part as well."
United and Celtic will now replay in Glasgow on March 18 for a semi-final against either Inverness Caledonian Thistle or Raith Rovers next month.
Before that, they will clash in the League Cup final at Hampden next Sunday.
Sunday's semi-final draw also threw up a tie between Championship rivals Hibernian and Falkirk.
Source : AFP
Source: AFP