The Saints, who no longer have the threat of the drop hanging over them following Fulham's relegation on Monday, went behind in the second minute when Christian Benteke out-muscled their defence to score with a thumping left-footed shot.
Ings, who had been out of action since tweaking a hamstring against Tottenham Hotspur in late April, levelled in the 19th minute, catching the Palace defence flat-footed as he controlled the ball before swivelling to fire home.
Nathan Redmond upended Wilfried Zaha to concede a penalty just before the break, but Southampton keeper Fraser Forster dived smartly to his left to keep out the resulting spot kick from Luka Milivojevic.
In the second half it was Southampton's turn to get off to a fast start and they scored two minutes in with a cleverly-worked corner from James Ward-Prowse that was played in to Che Adams, who steered the ball home from a tight angle.
With Palace floundering, Ings toe-poked his second in the 75th minute to lift Southampton to 14th spot in the table on 40 points, one behind 13th-placed Palace.