Matheus Franca’s stoppage-time goal for Palace denied Saints only a third Premier League win of the season and left them on the brink of an unwanted record.
Substitute Franca’s late header means Saints will be relegated on Sunday if they lose at Tottenham and Wolves beat Ipswich.
If relegation is confirmed this weekend, Southampton will be the first team to be consigned to the drop with seven matches still to play.
Saints did at least move on to double figures for the season, but they are still one shy of Derby’s record low total points tally of 11.
Juric remained proud of the performance despite feeling Saints did enough to claim all three points at St Mary's.
"I'm very proud of the way we played today and very disappointed that we didn't win because we did almost everything good," he said.
"I had a good sensation on the bench that we don't risk anything, that we are a compact group. I'm really sorry for the fans and for the guys that we didn't win.
"I think we deserved it. It was just a moment. I had a feeling that we are controlling the game.
"We didn't risk anything that they don't create opportunities or anything dangerous. I had a really good feeling, and then it was one moment.
"They were good with the cross, and we lost a little bit of height in this moment without Paul (Onuachu) and Flynn (Downes). Maybe this is the reason."
He continued: "I think that everybody needs a little bit of time to understand everything. I think the players made a step forward, and I played one step back.
"I had a very good feeling that we can compete today because there were moments that we pressed really well and stole the ball in the moment that we are waiting a little bit.
"I think we can do it better with the ball because in some situations we lose some balls in the middle. In the defensive way, we were really good.
"We deserve the win today. Like performance, in everything, I think we deserve it much more. We gained one point.
"We have to prepare well for the next game. I think if we play like this in every game, we will gain the points."