Goals from Lucia Leon, Katie O'Leary and Avilla Bergin put Spurs in the driving seat in the fourth round tie and secured a 3-0 win against their opponents from the FA Women's Premier League North.
The team has now reached the firth round for the first time since the mid-1990s.
"The first half performance had everything," said manager Karen Hills. "The girls were so determined to start well and every player stepped up to the plate.
"We opened the scoring after 11 minutes when Lucia Leon fired across the goalkeeper at the far post.
"The second goal was the pick that started with goalkeeper Toni-Anne Wayne then Jenna Schillaci found Sophie McLean down the left, she beat her full-back and crossed for Katie O'Leary to finish well.
"Lucia set-up the third goal on the half-hour, beat two players and delivered a cross for Avilla Bergin to tuck away.
"It was 3-0 at half-time and we knew as long as we were professional in the second half, that would be enough.
"Derby did put us under pressure in the first 15 minutes of the second half but I never felt like we looked like conceding a goal.
"Maya Vio, Wendy Martin and Ri Soobadoo controlled it in midfield and overall, it was a very good performance."
Spurs Ladies vs Derby: Toni-Anne Wayne; Lucia Leon, Alex Keown, Jenna Schillaci, Sophie McLean; Wendy Martin, Riana-Devi Soobadoo, Maya Vio; Bianca Baptiste (Laeh Rawle, 70), Katie O'Leary (Andra Marcoci, 75), Avilla Bergin (Leanne Mabey, 84).