Saints looked on course to securing just their third win of the Premier League season until Matheus França scored a late equaliser for Crystal Palace.
The 1-1 draw meant Saints have now dropped 25 points from winning positions, the most in the division, and could be relegated as early as Sunday.
Victory would have delayed their drop to the Championship but relegation will be confirmed if they lose to Spurs and Wolves beat Ipswich.
Some fans questioned whether this Saints squad, bottom of the table with 10 points, care, but Harwood-Bellis shot down that suggestion.
"I don't think it's a fact that the players don't care because that's just impossible," the one-time England international told the Daily Echo.
Southampton's Taylor Harwood-Bellis during the Premier League match between Southampton and Crystal Palace.Southampton's Taylor Harwood-Bellis during the Premier League match between Southampton and Crystal Palace. (Image: Stuart Martin) "I don't think you can play in the Premier League if you don't care. That just doesn't happen. There have been times when I could tell you we're not being good enough.
"That's a fact. Against a good opponent, you get punished. Not caring, we can put that to bed. That's just so far from what we're about.
"I thought we deserved more. It was difficult towards the end with the set-piece and when they're moving about so much.
"We'll look at the positives going on to Sunday and hopefully put a good performance in and give the fans something to cheer about again."
He added: "I've played in quite a few teams now, and I just can't put it down to spirit. I've played in teams where the spirit has not been together.
"We've got players who do care, who are running and training. We have no one who is throwing the towel in or anything like that.
Southampton's Taylor Harwood-Bellis and Kyle Walker-Peters battle for the ball.Southampton's Taylor Harwood-Bellis and Kyle Walker-Peters battle for the ball. (Image: Stuart Martin) "We come in, and we work every day, so I can't say it's spirit from the start of the season. It's the quality, and we've been punished this season."
Juric admitted Southampton's aim is to avoid becoming the worst team in Premier League history, and three points would have seen them achieve that target.
A win would have put them on 12 points, one more than Derby achieved in 2007/08, which remains a record low in the Premier League era.
"It's obviously in the back of your head, I wouldn't be lying if I said it wasn't. It's something that strives you on," Harwood-Bellis concluded.
"But it's not at the forefront of your mind. We've got eight games to go, and I'm pretty sure we'll pick up points towards the end of the season."