Despite shining in the Carling Cup this season with three Championship scalps, the Yorkshire side have not registered an FA Cup win for six years.
They crumbled again as last season's Conference champions romped to a richly deserved replay victory.
Rotherham saw an early appeal for a penalty waved away after Reuben Reid appeared to be held back in the area.
Richie Barker might have done better but fired over after spinning cleverly and a Reid shot was easily saved before Barker headed over.
But it was Aldershot who made the breakthrough after 15 minutes through Hudson.
Hudson couldn't believe his luck when Pablo Mills made his dreadful gaffe and passed the ball to Hudson, who tore forward into the area and lashed a fierce shot to beat stand-in keeper Steven Cann.
The setback knocked the stuffing out of the home side and settled, Aldershot who might have stretched their lead before the break.
Two minutes after the opener Marvin Morgan turned and shot just wide after impressive work from Anthony Straker.
Another effort was from Morgan was well saved by Cann, but Rotherham were fortunate not to go further behind as John Grant smashed a 25-yard rising shot over the bar after he breezed past Mills.
Despite grabbing large slices of possession Rotherham had a rare chance of an equaliser but a Reid header looped over after a corner from Alex Rhodes.
But then as Rotherham pushed forward looking for an equaliser they were caught on the break as Aldershot made a lightning break ten minutes into the second half.
Straker found Hudson and as Rotherham looked for an offside flag he fed Morgan, who had space and time to place a firm shot out of Cann's reach.
Morgan came close to his second after 72 minutes as Rotherham threatened to crumble, but Cann, in his first match of the season, dived well to hang on to the shot from 20 yards.
Ten minutes from the end Hudson added the third when he drilled in a fierce shot from just inside the area.