Huddersfield Town have been relegated with six games to go, equalling an unwanted Premier League record set by Derby County in March 2008.
The Terriers' defeat, combined with victories for Burnley and Southampton, confirmed the visitors' demotion back to the Championship.
Milivojevic sent Ben Hamer the wrong way from the penalty spot in the second half after Wilfried Zaha had been fouled by Juninho Bacuna.
Patrick van Aanholt's composed finish secured a valuable victory for Palace, who move eight points clear of the bottom three.
Jan Siewert's side spurned several chances to open the scoring in the first half, with Chris Lowe twice forcing fine saves from Palace goalkeeper Vicente Guaita.
Karlan Grant also missed an excellent opportunity with the score at 0-0, firing wide of the target with only Guaita to beat.
