Goals from Alejandro Dominguez and Arsenal loanee Joel Campbell left United with a huge mountain to climb to make it into the quarter-finals of the Champions League.
Van Persie wasted a good chance to score a vital away goal at the end of the match as the Dutchman put in an uncharacteristically poor display.
The former Arsenal striker, speaking to Dutch broadcaster NOS after the match, said that his cause was not helped by the positioning of some of his team-mates.
"It was quite difficult because my team-mates are often in the area where I want to play," the United striker said.
"That's why I have to change my tactics. Unfortunately, they are often in my area.
"It was much too slow today and we only had one shot on target in the whole match against Olympiacos, which is far too little.
"Everyone is very disappointed."
United have to win by three clear goals to be sure of their passage to the last eight of the competition they were claiming they could win in the run-up to Tuesday's first leg in Athens.
With United out of both domestic cup competitions and 11 points adrift of the top four in the Barclays Premier League, the Champions League is the only trophy David Moyes' team can win this year, and Van Persie is refusing to throw in the towel just yet.
"It is still possible, because with all due respect to Olympiacos, we should be able to win 3-0 at home," the striker added.
"It is not that they were fantastic (in the first leg), we just mostly failed."
Former United captain Roy Keane hit out at the team following the defeat.
He told ITV 1 the Red Devils lacked confidence and quality, but Moyes disagrees and the Scot insists United can still progress following the second leg, which takes place in three weeks' time.
"I think there is undoubtedly talent at Manchester United but tonight we didn't show it," the United manager said.
"Me and the team we didn't show it together.
"It is something we will have to do. We will put it right. We are determined to put it right and will have opportunities to do that in the coming weeks.
"There is a second game to come and Old Trafford has seen some great night s in the past and I am looking forward to hopefully seeing another one."