Jenny Moore converted Avilla Bergin's corner for 1-0 on 20 minutes and five minutes later it was 2-0 as Mollie Kmita cut the ball back for Maya Vio to dispatch into the top corner.
It was 3-0 before the break when Leah Rawle released Lucia Leon down the left, she hit the byline before picking out Bergin to finish from 12 yards. Rawle made it 4-0 on 65 minutes when she finished one-on-one after more good work from Leon. Plymouth score a consolation in injury time.
"It was a good team performance," said manager Karen Hills. "We were 3-0 up before half-time, controlled the game and defensively, we handled anything that came our way.
"I must give e mention to Maya Vio, who scored a fantastic goal and Jenny Moore, who made a nuisance of herself up front again and got us underway with the first goal.
"The win moves us into the last 16 and as I've already said, this is a big competition for us and one we want to do well in.
"It was the perfect start to a busy week for us."
That week continues with another cup game on Thursday - this time against neighbours Enfield Town in the Ryman League Cup at Cheshunt FC on Thursday night (7.45pm) - before the long trip to Cardiff in the FA Women's Premier League on Sunday.
Spurs Ladies have been drawn at home to Leyton Ladies in the London Cup on Sunday, October 18. The FAWPL match at West Ham will be re-arranged.
Spurs Ladies v Plymouth: Toni-Anne Wayne, Mollie Kmita, Alex Keown, Jenna Schillaci (c), Lucia Leon, Amy Cooper, Katie O'Leary (Riana Soobadoo 46), Maya Vio (Shannon Moloney 68), Leah Rawle, Jenny Moore, Avilla Bergin (Rosie Kmita 80). Substitutes (unused): Bianca Baptiste, Chloe Morgan (gk).