The former England captain, 33, took the first penalty after the match had finished 2-2 following extra time, but his shot was saved by Zack Steffen.
Luciano Acosta and Nick DeLeon also missed from the spot as DC lost the shootout 3-2.
Earlier, DeLeon had scored in the 116th minute to send the game to penalties.
Frederic Brillant had given DC the lead in the 21st minute, before Federico Higuain – the brother of AC Milan striker Gonzalo Higuain – scored twice for Crew.
“It’s disappointing to lose on penalties but as a team we can be extremely proud of what we have achieved in the last few months,” said Rooney.
“Really we had no right to make the play-offs. To do that we have brought belief back to the fans, the team and we have to build on that.”
The team were bottom of the Eastern Conference when Rooney, Manchester United’s record scorer, joined from Everton in June.
Rooney, who was made captain after three games, scored 12 goals and provided seven assists as DC United clinched a play-off spot with a game to spare.
Elsewhere, Los Angeles FC’s maiden MLS season ended in defeat as they lost 3-2 to Real Salt Lake, with ex-Manchester City defender Nedum Onuoha playing the full game for Real.
There will be plenty of British interest in the conference semi-finals, with Onuoha, New York Red Bulls’ Bradley Wright-Phillips and Scotland international Johnny Russell of Sporting Kansas all involved.
The conference semi-finals are played over two legs between 4 and 11 November.
Ties: Columbus Crew v New York Red Bulls, New York City v Atlanta United, Portland Timbers v Seattle Sounders, Real Salt Lake v Sporting Kansas.