De Gea's rollercoaster campaign saw him initially struggle to win over United fans, but improved performances could not help the club as they finished behind rivals Manchester City on goal difference.
"Maybe it was hard to follow in my footsteps. It can happen. People got compared to Peter Schmeichel, now maybe it's me," Van der Sar said.
"De Gea has to ignore that and just play his own game."
The 41-year-old won four league titles with United, taking the Golden Glove award for a record 21 clean sheets in 2008-09.
De Gea has improved as the season has gone on, and was recently named in Spain's provisional squad for the European Championships.Van der Sar continued: "He did get criticism last season but I made mistakes at Ajax when I was young, and also at Juventus and United.
"The main thing is how you recover in the next game, or the game itself. He did that. He became more important as the season went on and made good saves."
Van der Sar has kept the club close to his heart following his retirement, and understands the disappointment that his former teammates would have felt on the final day.
He believes that De Gea and the rest of the United squad have done well under the circumstances, however, having lost key players through injury.
He added: "Whilst I don't think they anticipated becoming champions, the disappointment at those final two or three minutes would have been great. To lose it in the last seconds must have hurt so much.
"If you look at it, they did quite well. They got more points than last year and kept a lot of clean sheets, even with Nemanja Vidic out for so long."