Huddersfield, who have propped up the table since December, join Derby County and Ipswich Town as the only teams in Premier League history to be relegated with six games left to play.
Milivojevic opened the scoring in the second half, sending Ben Hamer the wrong way from the penalty spot 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.
Huddersfield spurned several chances to open the scoring, but they failed to make the most of those opportunities as their two-year stay in the Premier League came to an end.
Having started the day 16 points adrift of safety, it was always going to take a remarkable turnaround for Huddersfield to preserve their top-flight status for a third season.
The Terriers’ energetic and fearless approach under David Wagner last term endeared them to neutrals as they achieved what many regarded as impossible by avoiding an immediate return to the second tier.
Life has been much more difficult this season.
Wagner left the club by mutual consent in January with the club eight points adrift at the foot of the table having taken only 11 points from 22 games.
Siewert has been unable to spark a recovery, losing eight of his nine league games in charge with the sole exception a win at home against Wolves in February.
One of Town’s main problems has been their lack of goals. They have scored only 18 in their 32 games, finding the net just once away from home in 2019.
They were unfortunate not to make the scoresheet at Selhurst Park, with Chris Lowe twice forcing fine saves from Vicente Guaita.
The Spanish goalkeeper produced a one-handed save to prevent the defender’s header from nestling in the back of the net, before tipping another goal-bound effort past the far post.
Karlan Grant also went close to giving the visitors the lead before half-time, but the former Charlton man failed to direct his effort on target.
The Huddersfield players received a warm reception from the travelling Terriers fans at the end of the game, but Burnley’s win over Wolves and Southampton’s victory at Brighton mean Siewert’s side will be playing Championship football again next season.
