McGoldrick's 25-yard thunderbolt midway through the second half, his 13th goal of the season, lifted Mark Wotte's administration-haunted side to within two points of safety.
The match took a while to spark into life and neither goalkeeper had a save to make until the 22nd minute.
Palace keeper Julian Speroni showed superb reactions to palm away Bradley Wright-Phillips' powerful shot area, after the former Manchester City striker had cut in from the right.
Wright-Phillips posed a constant threat in a first half dominated by the home side and he played a major part in Palace captain Clint Hill being forced to replace the stricken Speroni in goal ten minutes before the interval.
Speroni came flying across his line to keep out Wright-Phillips' low left-foot shot, before diving bravely at the feet of Andrew Surman to prevent the Saints midfielder tapping in the rebound.
Surman caught Speroni in the face as he went to shoot and the Argentine made his way down the touchline with blood pouring from his nose. With no replacement keeper on the bench, centre-back Hill donned the gloves, reducing the Eagles to ten men.
Speroni re-emerged for the final minute of the half after being patched up by Palace's medical team, by which time Hill had shown he was an able stand-in by saving well at the feet of McGoldrick.
Wright-Phillips should have scored when he was sent one-on-one with Speroni by Jason Euell's through ball, but shot straight into the arms of the dazed keeper.
Speroni continued to take centre stage and produced another brilliant fingertip save to turn Adam Lallana's angled shot behind for a corner in first-half stoppage time.
It was always going to take something special to beat Speroni and it arrived in the 67th minute when McGoldrick collected Surman's pass, before running at a retreating Palace defence and firing into the top right corner with an unstoppable long-range shot.