The Hillsborough side, who will be happy to settle for a mid-table finish, looked to have snatched all three points through centre-back Richard Hinds' close-range header only for de Vries to conjure up a stunning save that salvaged a point for the Swans.
Wednesday, whose miserable run of two points from four games had ended their own play-off hopes, pressed the Welsh side back in the opening minutes, Francis Jeffers almost finding a way through after linking with midfielder Sean McAllister.
Marcus Tudgay carved out a half-chance for himself when he cut inside from the left flank and shrugged off full-back Angel Rangel's weak challenge only to drag his low right-foot shot the wrong side of the near post.
Swansea found a way through after Jermaine Johnson blundered as Darren Pratley raced into the area only to be stopped by Lee Grant, the Wednesday keeper diving bravely at his feet.
Johnson made amends in the 21st minute with a last-ditch tackle on Rangel in the area as the Spanish defender was about to shoot.
Swansea, content to hit the Hillsborough side on the break, shredded Wednesday back in the 37th minute, Pratley passing inside to the incoming Jordi Gomez who hit the ball on the run only for his low shot to be smothered on the line by Grant.
On-loan striker Luke Varney, signed from Derby just 24 hours earlier and on for the injured McAllister, should have snatched the lead seconds before the break after a Jeffers flick-on, but his 16-yard shot crashed against the legs of de Vries.
Centre-back Gary Monk headed a Mark Gower free-kick wide of the far post as the Welsh side probed for the opener. It almost came from a curling Gomes free-kick from 20 yards that was pushed away by the diving Grant.
De Vries performed heroics with ten minutes to go after Swansea had failed to cut out a Darren Potter free-kick.
Tudgay picked out Hinds whose close-range header was tipped away one-handed by the keeper and, as Tudgay followed up with a low strike, de Vries reacted quickly to snatch the ball on the line.