Smith picked up a back-pass which resulted in Michael Carrick giving United the lead before the Premier League giants made sure of victory when Marcus Rashford capitalised on indecision from the goalkeeper on the edge of the penalty area.
The 42-year-old is quoted by BBC Sport as saying: "I'm really proud of the players, we had to be disciplined. They've had a tough week by their standards but we knew we had to be strong. When you give away soft goals you will get punished.
"[Smith] was outstanding last season and he'll bounce back from this. He's a strong character, we've all had moments like that as players."
United have been drawn against neighbours Manchester City in the last 16 of the competition.