Keanan Bennetts, Kazaiah Sterling, Ryan Loft and Samuel Shashoua were on target in a one-sided third round encounter at the Lamex Stadium.
Rotherham, who play in the North-East Division of the Football League Youth Alliance, set up in a 4-2-3-1 system, sat deep and looked to use the pace of Kudakwashe Muskwe and Darnelle Bailey-King on the break.
That allowed us to enjoy the lion's share of possession and the likes of Bennetts, Sterling, Shashoua and Marcus Edwards, backed-up by Zenon Stylianides and Charlie Owens, were always going to take advantage.
The team's attitude was spot-on from the start although there was a scare when Alfie Whiteman was forced into a fine early reaction save to tip over Bradley Peace-Macdonald's header. That was one of only two clear-cut chances the visitors created all night.
We took the lead on 13 minutes as Owens picked out an inch-perfect pass for Bennetts, dissecting Rotherham's defence in the inside right channel. Bennetts' first effort struck the inside of the post but the winger was alert to the rebound and converted from close range.
Rotherham put up the barricades as the pressure increased and managed to block goalbound efforts from Sterling twice and then Owens before springing out themselves as Muskwe got away down the right and volleyed just wide.
Below: Japhet Tanganga takes on Brandon Potts
Joe Muscatt replaced Nick Tsaroulla at half-time and the second half was all Spurs.
Edwards fired straight at Laurence Bilboe from 25 yards and the goalkeeper did well to divert Bennetts' shot down to his right before touching the ball away from Shashoua following in.
Either side of the hour mark Shashoua shot straight at Bilboe and Owens' free-kick grazed the roof of the net.
You sensed the second goal was coming and it duly arrived on 65 minutes. It stemmed from quick feet and a burst of pace from Stylianides down the left before cutting the ball back for Sterling, whose shot found the net via a deflection.
It was soon 3-0 as Edwards picked out substitute Ryan Loft and the big striker made it look easy, controlling before side-footing home from 14 yards.
Shashoua got his rewards for a bright display on 83 minutes when his 25-yarder took a slight deflection to leave Bilboe helpless.
There was still time for Bilboe to deny Shashoua again and Jaden Brown to shoot over after a venture forward. All in all, a job well done.
Spurs: Whiteman, Tanganga, Tsaroulla (Muscatt 46), Owens, Carter-Vickers (c), Brown, Bennetts (Duncan 79), Stylianides, Sterling (Loft 71), Edwards, Shashoua. Substitutes (not used): McDermott (gk), Marsh.
Rotherham: Bilboe, Bailey (c), Smalley, Dickinson, Peace-Macdonald, Rose, Bailey-King, Potts (Abraham 63), Muskwe, Wiles (Adeyemi 46), Redmayne (Saxton 88). Substitutes (not used): Fidler (gk), Martin.