An outstanding hat-trick from Belgium youth international Ismail Azzaoui proved to be the difference between the two sides as Spurs ran out 4-1 winners at Victoria Road.
Rovers found themselves a goal down after just 90 seconds as Azzaoui opened the scoring, but the Lancastrians responded well as Sam Joel levelled five minutes later.
Tottenham captain Joshua Onomah regained the lead for the hosts and it was one that they wouldn't relinquish as two superb second half goals from Azzaoui put the game beyond Rovers and Spurs' name in the hat for the fourth round.
Rovers were able to name a much more experienced side than had featured in the league this season as development squad regulars Jack Doyle, Sam Lavelle, Connor Mahoney, Hyuga Tanner and Modou Cham were all named in the starting XI for the match at Dagenham.
But despite this it was a sloppy start for the young Rovers as they fell behind after just 90 seconds.
Hesitancy in possession proved costly as Lavelle was caught by Shayon Harrison and he fed Ismail Azzaoui who beat Andrew Fisher in the Rovers goal with a curling shot to put the home side in front.
Rovers needed a response and it came just five minutes later. Mahoney sent over a free-kick from the right, Lavelle stole in only to see his shot slam the cross bar, but Under-18s top scorer Sam Joel was on hand to net his 9th of the season.
There was little between the two sides for long periods of the first half. Spurs captain Joshua Onomah fired across the face of goal, whilst Modou Cham saw an opportunity wasted when he blazed over from the edge of the box.
However a goal just before the half hour would ensure it was the home side who would lead at the break. A ball into the box from the right by Anthony Georgiou saw Onomah let fly from close-range. Fisher in the Rovers goal made a superb point-blank save, but was powerless to prevent the midfielder from slotting home the rebound.
But just as the first half, Rovers were sloppy starters as play resumed for the second. From their own kick-off Rovers were caught in possession, Azzaoui taking full advantage with a superb jinking run and finish for his second of the match to put daylight between the two sides.
Only the fingertips of Fisher prevented Spurs from grabbing a fourth on 53 minutes. Getting the slightest of touches to a Harrison shot to divert the ball onto the post and away to safety.
Luke Wall and Lewis Mansell were thrust on for Rovers as they looked for a way back and Mansell's first touch very nearly resulted in a second goal for the visitors but he somehow headed Mahoney's centre wide of target with the goal at his mercy.
To their credit Rovers continued to press, but long range efforts from George Pierce and Connor Mahoney were never going to trouble the Spurs back line.
With bodies thrust further forward Rovers were caught on the break with five minutes to go. Man-of-the-match Azzaoui capping a fine display with a sumptuous third goal on the night to ensure it would be Tottenham's name in the hat for the fourth round draw.
TOTTENHAM: Glover, Walker-Peters, Muscatt, Amos, Carter-Vickers, Walkes, Georgiou (sub Loft 81 mins), Onomah (sub Stylianides 77 mins), Harrison, Goddard (sub Ross 88 mins), Azzaoui.
Subs not used: Voss (GK), Sterling.
Goals: Azzaoui 2, 46, 85 mins, Onomah 29 mins.
ROVERS: Fisher, Nyambe, Doyle (c), Pierce, Lavelle, Edgar, Mahoney, Hardcastle (sub Mansell 69 mins), Joel, Tanner, Cham (sub Wall 65 mins).
Subs: Williams (GK), Edo, Platt.
Goals: Joel 6 mins.