The Hoops suffered a surprise 2-1 home defeat to Aberdeen last Saturday as they crashed out of the William Hill Scottish Cup.
The unbeaten Scottish Premiership leaders had already suffered a shock slip up in the League Cup earlier this term when they were beaten 1-0 at Parkhead by Championship outfit Morton.
But Scotland utility man Mulgrew claims reaching the Champions League group stages to take on European giants Barcelona, AC Milan and Ajax and a record run of clean sheets that now stretches to 11 league games is evidence of a worthwhile season.
He said: "No, I don't think the season has been a failure. Let's not forget that we played in the Champions League this year, which is a massive achievement for this club.
"I know we qualified for the last 16 the year before but this time it was always going to be difficult with the group we were in. We believed we could do it but it wasn't to be in the end.
"But you have got to see it as an achievement because there is still three qualifiers to get there.
"Fair enough, we have had disappointments in the cups but our league form has been very good. I think it's been a good season overall."
Boss Neil Lennon reacted to his side's Dons loss by demanding they finish the job and complete their 38-match league programme without defeat.
The next leg of that challenge will come at home to St Johnstone on Sunday afternoon but Mulgrew insists the Celtic players are focusing on their Perth opponents rather than their unbeaten run.
"We're not speaking about that too much," said the 27-year-old. "It's hard to ignore the fact we could go the season unbeaten is a possibility because people keep mentioning it.
"But we are just taking each game as it comes and hopefully we can win the next one against St Johnstone.
"Maybe we need something to motivate us and to keep us going and the unbeaten run can do that for us. But at a club like Celtic, there is always pressure on us, even in friendlies.
"The manager would never let us get complacent. He keeps on top of us and demands high standards."