The defeat dented Rotherham's chances of avoiding relegation while Barnsley now stand on the brink of securing a play-off place.
Inside the opening minute, Cauley Woodrow's header was turned over by Viktor Johansson and Morris scored from the resulting corner, despite appearing to impede the keeper as he jumped to connect with the ball.
After lengthy treatment, Johansson was replaced by Jamal Blackman.
There was a great chance for the visitors when Michael Smith delayed his shot, allowing Callum Brittain to make a vital block.
Freddie Ladapo wasted two great opportunities and Chiedozie Ogbene had a shot saved.
Daryl Dike had a chance to double Barnsley's lead, but Richard Wood blocked the shot.
Further chances for Rotherham went begging when Smith headed wide and Ogbene put a close-range effort straight at Collins two minutes from time.
Warne said: "I thought we'd done enough to get a win, let alone a draw. The lads gave everything but unfortunately the most crucial part of that is putting the ball in the net and we didn't today, although we created enough chances.
"I'm a tad frustrated. I think everybody had the same opinion as me (about the goal) but the only one that counts is the ref's.
"Goalkeepers get protected more than any other player on the pitch and I've had a player sent-off recently for doing less than that and having a three-match ban.
"That's the only goal of the game and my keeper has to leave the pitch after one minute and he's got what looks like a broken eye socket.
"I'm not saying there's any maliciousness from the player at all, but I've spoken to the player and he says that he thought the ref was going to blow straight away. He didn't think it was going to stand and nor did anybody else but unfortunately, whichever angle the ref had, he believed that it was a fair and legitimate attempt at the ball, which surprised me.
"I tried to speak to the ref - he didn't want to speak to me - but what difference does it make?
"However, we did have 94 more minutes to get ourselves back in the game so I can't hang my hat on that.
Barnsley head coach Valerien Ismael said: "It was a really, really poor performance. We got the three points, which I think is the most important thing.
"Big congratulations to Rotherham - they deserved more than losing the game.
"It was the expected fight. Maybe we scored too early. We never came into the game and we were always too late.
"Our decision-making was too slow. At the end of the day, in two weeks, nobody will speak about that performance. It's all about getting the three points and we did it.
"When you are in a season when everything goes in the right direction, you win such a game.
"I didn't see the situation from my angle (for the goal) but at the end of the day it doesn't matter now, we have the three points.
"We are not over the line just now. We will see if it's enough or if we have to get the next point."