The Scottish champions confirmed their place in Friday's draw in Nyon with a goalless 90 minutes against Elfsborg in Boras on Wednesday night.
Kris Commons' goal from the first leg in Glasgow ultimately proved to be the difference between the two sides and left the Hoops with the prospect of facing Austria Vienna, Slovenian side Maribor, Shakther Karagandy of Kazakhstan, Bulgarians Ludogorets Razgrad or Legia Warsaw (Poland).
However, Mulgrew is not unduly worried about any of their possible opponents.
"We know what to expect in the next round, in terms of how tough it will be," he told the Evening Times.
"We are just waiting to see who we get. Whoever it is, we will be ready."
Neil Lennon's men are within two games of reaching the group stages of European football's elite club competition for the second successive year, and even if they lose out they will be guaranteed a place in the Europa League.
Mulgrew was just glad to get through a tricky tie against the Swedish champions.
He said: "We knew it was going to be a tough match against Elfsborg after beating them 1-0 in Glasgow. It was just about getting the result, and we did that.
"Although we knew it would be enough to see us through, we never set out to try and get a 0-0. That's too dangerous.
"We came to win and went looking for a goal and we had good chances, especially when Joe Ledley was in and was pulled back as he shot.
"We never managed to get the win, but, thankfully we did manage to keep a clean sheet."
Neil Lennon hopes to add at least one new player before the play-off ties take place on August 20/21 and 27/28 but in terms of numbers, that will only make up for the imminent departure of Kelvin Wilson.
The Englishman will return to Nottingham Forrest for a fee reported to be £2.5million.
Mulgrew was not surprised that the 27-year-old defender ignored the mounting speculation about his future to turn in another good performance in Sweden.
"Kelv was great," he said. "He is a good professional with a great attitude, so we knew he'd do a job for us."
Mulgrew had to come off in the second half with a knee injury which will be assessed on Thursday ahead of Celtic's match against Liverpool in the Dublin Decider on Saturday.
He said: "The knee locked up and I couldn't run or put any weight on it. I will have to see how it is tomorrow [Friday]."