Veerle Buurman's stunning late strike sealed the victory with the 19-year-old firing home with a fierce effort that beat Spurs goalkeeper Lize Kop in the 86th minute.
Eveliina Summanen's cross earlier deceived Hannah Hampton to get the visitors back on level terms seven minutes into the second half, after Kerr's header had put the Blues ahead five minutes before the interval.
The win, which also featured Kerr's Australia team-mate Ellie Carpenter, ensured Sonia Bompastor's side still have a chance of rescuing a disappointing season in the wake of their midweek UWCL exit to Arsenal, as they now target a seventh Cup title win.
'We will fight until the end to have as many titles as we can,' Bompastor told the BBC.
"The FA Cup is part of the best memory I had in my whole career as a player and manager, being able to go to Wembley and play a final there.
"It's such a special moment. We'll see the draw and who we have to play but will do everything we can to reach the final again."
Australia's Clare Hunt was unavailable for Spurs through injury, while Charli Grant did not feature as she makes her way back from an injury that saw her miss the AFC Women's Asian Cup.
Manchester City, featuring another Matildas star in second-half substitute Mary Fowler, are also into the semi-finals after Khadija Shaw's goal earned them a narrow 1-0 win away against Championship side Birmingham.
The goal came after only eight minutes. Iman Beney played a fine ball in behind and Shaw reacted quickest to slot past goalkeeper Lucy Thomas.
Thereafter the leaders of the second tier fought bravely to stay in the game, though it was City that came closest to scoring a second, Thomas brilliantly turning the ball against the bar in the second half from Yui Hasegawa's stinging shot.
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