The result has led to speculation head coach Unai Emery will be sacked at the end of the season, given the investment in Neymar and Kylian Mbappe last year that will cost the club a total of €400million in transfer fees when the latter's move from Monaco is finalised.
Neymar's future has also come under scrutiny, with a number of reports claiming he could return to Spain by joining Madrid, and Di Maria admits there is likely to be significant upheaval before next season begins.
"It was a tough blow," he told Ole when asked about the Champions League exit. "We spent €400 million on players, the club had a different expectation with Neymar and it was normal that that expectation was very high.
"The club really felt it to be knocked out in the round of 16. There is going to be a big change for next year.
"It was obvious that, because of the arrival of important players, the fans were going to criticise and be disappointed. But you know you have to lose and lift your head when it happens.
"We now have three competitions ahead and we'll try to win them all. If we do, it would be really positive, because we couldn't do it last year."
The midfielder is relishing being able to play alongside Lionel Messi once more – something he says will go down as the pinnacle of his career.
"I love playing with 'The Dwarf'," he said. "It's amazing, unforgettable. It's something I'll be able to tell my daughters: I was there, I shared a pitch with him. It was the best thing that happened to me in football."
Di Maria also feels Messi deserves to win the World Cup this year, given what he has achieved, although he accepts their struggle in qualification means Argentina might not be among the favourites.
"There is no doubt football owes Leo a World Cup, for everything he's done and achieved in his life. Of course, we want to help him," he said.
"It's possible we're not a contender because of how we qualified, or because of how we were playing before Sampaoli arrived. But having Messi gives you a boost.
"We have players who are at the best clubs. That makes you a contender. I'd like to be remembered for winning a World Cup.
"The pressure [to qualify] was huge. We could have missed out, the cycle of our best could have come to an end... it was hard to accept. But, with the qualification, we've been liberated. Not just me, but Leo, everyone."