The hosts took the lead in the 19th minute when a cross-shot from Olufela Olomola came back off the bar and Sam Surridge followed up to fire past Glenn Morris.
Yeovil then received the first of their red cards when Connor Smith was shown a second yellow and on 44 minutes the scores were level when Mark Randall sent a looping volley into the far corner.
The drama was not over yet as in the third minute of first-half stoppage time, Omar Sowunmi was given a straight red for bringing down Jordan Roberts just outside the box.
Despite their numerical advantage, Crawley struggled to break the Glovers down, before the game took a bizarre turn and was temporarily suspended on 80 minutes and the players went off as a drone flew over the pitch.
When play resumed 10 minutes later, the visitors did eventually make the breakthrough in the 87th minute when substitute Thomas Verheydt headed in a Jimmy Smith cross.
Southampton manager Mark Hughes was delighted to see his new side reach the FA Cup semi-finals but also warned that the Premier League remains a concern.
Borussia Dortmund winger Andre Schurrle was delighted with the team's hard-fought 1-0 Bundesliga victory over Hannover 96 on Sunday afternoon.
Chelsea defender Andreas Christensen has revealed that he sought advice from club legend John Terry after making a few costly mistakes recently.