A second half goal spree took them to a 5-1 victory against Preston North End of the FA Women's Premier League North.
Spurs will now travel to Derby County Ladies - from the same league as Preston NE - in the fourth round in a tie scheduled for Sunday, March 8.
"It's a special competition and this is a special achievement for the players," said manager Karen Hills, who won the trophy during her playing career at Charlton.
"We now face an away tie at Derby but the girls are confident and we're looking strong at the moment."
The teams traded chances in the early stages of Sunday's third round tie played at New River Stadium, Tottenham.
Katie O'Leary and Lucia Leon went close, PNE had a goal disallowed and Avilla Bergin was denied by the PNE goalkeeper before Maya Vio opened the scoring with a sweet strike from the edge of the area just before half-time.
The ladies stepped up a gear in the second half and three goals in the space of 13 minutes put the tie out of PNE's reach.
Katie drilled in number two on 56 minutes, Wendy Martin tapped in from Sophie McLean's cross for 3-0 and Maya capped a fine display when she latched onto Avilla's backheel and finished a team move for 4-0 on 69 minutes.
PNE pulled a goal back when Sarah Chadwick was there to convert the rebound from a Toni-Anne Wayne save but Spurs had the final word when Lucia fired home for 5-1 in the final minute.
"We dominated the second half and every player was exceptional," added Karen. "We passed the ball well, pressed, the midfield three looked so strong and the front three created chances.
"We've come a long way this season and the way we're playing at the moment we're looking forward to every game."
The ladies are waiting for a rescheduled date for their Ryman Cup semi-final against Gillingham so next up, Keynsham in the FA Women's Premier League South at Ware FC on Sunday.
Spurs Ladies vs Preston North End: Toni-Anne Wayne; Lucia Leon, Jenna Schillaci, Alex Keown, Sophie McLean; Shannon Maloney, Wendy Martin, Maya Vio (Andra Marcoci, 75); Avilla Bergin, Katie O'Leary, Bianca Baptiste (Riana-Devi Soobadoo, 75). Unused sub: Leah Rawle.
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