Fulham appeared to be heading for their first FA Cup semi-final appearance since 2002 after Aleksandar Mitrovic opened the scoring in the 50th minute.
But goals from Bruno Fernandes and Sabitzer earned United a 3-1 win after Fulham's Willian and Mitrovic were sent off.
"I think we played a good first half. We had some chances, missed it. But after the half time, we came out and conceded a goal and that is not what we wanted," Sabitzer told MUTV.
"So we know at this time it was going to be tough and hard. But, we still believed it and we had good team spirit and mentality that you can always see on the pitch.
"To be fair, I have to say to be two men down for Fulham, it wasn't easy
''(we have) a good group of guys. You can see and feel it in the dressing room, on the pitch, off the pitch. We worked together, close. We stick together all the time."
Sabitzer, who joined on loan from Bayern Munich in January, said scoring his first goal for United left him feeling "overwhelmed".
"I was looking for it, scoring the first at Old Trafford in front of these amazing fans," the Austrian added. "And that important one, at 2-1 it was a massive one"
United, who are third in the Premier League, next travel to fifth-placed Newcastle United on April 2.
