United also lost striker Romelu Lukaku to a serious-looking head injury as they were forced to settle for a point for a third game in succession.
The hosts had a strong penalty appeal turned down and had a late Paul Pogba goal disallowed for offside, but Saints arguably deserved a point for their resilience.
United boss Jose Mourinho said: "Craig Pawson is a fantastic referee, one of the most promising referees in Europe, and it was bad for him. It was one of the key points. The other key points were easy goals that we missed.
"Another key point was David De Gea's save on the Shane Long shot. But the clear, clear point, especially with a result at 0-0, is a clear penalty that was not given. I watched on the touchline and it looked to me very clear, but it's 50 metres distance, so I gave the referee the benefit of the doubt. I know my players and, when I see players like Juan Mata almost in hysterics claiming, I could understand immediately - no doubt.
"And then, at half-time, I watch it and it is a penalty like Marcus Rashford at Leicester was a penalty, like Ander Herrera against Manchester City was a penalty and it is unlucky because honestly I told you that and I repeat without a problem, that the three referee performances, Michael Oliver against City, Jon Moss against Leicester and Craig tonight were good referees with good performances, but unlucky decisions for us that punished us."