United led 2-0 after 11 minutes, through Javier Hernandez and the recalled Wayne Rooney, but Chelsea roared back in the second half.
Goals from substitute Eden Hazard and Ramires levelled the match at 2-2 and the inspirational Mata was only denied a late winner by fellow Spaniard David de Gea's superb reaction save.
"We were playing in a stadium like Old Trafford and it was Man United, so it could be a good result," Mata told ITV1.
"But in the second half we had so many chances to score. De Gea was good, saving some of them.
"He saved very well (from me), with a foot I think, and Oscar had so many chances.
"It's a day to be proud of the team. (United) scored two in the first minutes but we tried very hard and came back.
"We have a replay at home, we are strong at Stamford Bridge with our supporters, and we will try to get through."
Blues boss Rafael Benitez felt his side should ultimately have won the game, such was their second-half improvement.
"I think so," he told ITV1. "The second half we were on top and controlling the game."
Benitez was seen embracing compatriot De Gea after the game and he said: "(Mata) told me De Gea made a great save, so it was unlucky but it was a great save."