Agudelo flicked a back-heel pass to Lee Nguyen, who then returned it to Agudelo for a game-deciding volley into the net.
Matt Reis made four saves for the Revs (14-11-9), who stretched their winning streak to three games and made the playoffs for the first time since 2009.
Andy Gruenbaum recorded one save for the Crew, who finished 2013 with a 12-17-5 record.
2014: NFL Cheerleaders [PHOTOS]
Arizona Cardinals player formally charged