United took the lead in the first hald through a wonderful strike from Michael Carrick. Rooney had an indirect free-kick blocked by the Northampton wall and the ball fell to the midfielder, who placed his shot into the top corner from just outside the area.
However, despite finding themselves behind, Northampton responded well and almost found an equaliser when Kenji Gorre had an effort tipped onto the bar from Sergio Romero.
The Cobblers then found themselves level just before half-time. Daley Blind brought down Sam Hoskins in the area and Alex Revell stepped up to fire his penalty past Romero.
The match stayed level until the 68th minute, and it was Rashford once again who played a part in dragging the Red Devils out of trouble. The England international assisted Ander Herrera, who fired home a low effort.
Rashford then combined once again with Herrera, but this time it was the former who assisted Rashford. The Spaniard fired a long ball forward which was looked like it would be dealt with by Northampton 'keeper Adam Smith, but Rashford harried the 'keeper and made him take a fresh air shot at his clearance, leaving the youngster with an open goal. Rashford subsequently blasted the ball home, avoiding what would have been the miss of the season.
United reach the last-16, but they still haven't reached their best.