Otis Khan opened the scoring from the spot, an incident which saw Cristian Montano sent off for the Valiants.
After the interval, James Gibbons joined Montano for an early bath seeing red for a nasty tackle on Ryan Dickson before Marcus Harness set up an additional 30 minutes with a wonder goal.
Khan then fired in an excellent half-volley with help from the post but an Antony Kay dipper from 25 yards restored parity once more.
However, Vale were level for only a minute as Zoko poked home to secure Yeovil a home tie against Bradford City.
Manager Darren Way made two alterations to his midfield following Saturday’s defeat to Lincoln City.
Jake Gray was handed a first start in four matches with Connor Smith replacing the injured Oscar Gobern.
Elsewhere, Sam Surridge partnered Francois Zoko up front while Dickson recovered from a neck injury to slot in at left-back.
Town has all the chances in the opening 10 minutes, and looked threatening when direct and on the counter.
Khan’s free kick on two minutes flashed across the area, just out of reach of a stretching Omar Sowunmi before an excellent defensive challenge from Tom James on Montano led to the hosts’ next attack.
After Sowunmi cleared the resulting corner, Khan drove the Glovers into a three-on-two situation.
With Connor Smith to his left and Gray to his right, the tricky winger opted to play in the other but his overhit pass led to ex-Luton Town midfielder finding the away terrace.
Former Yeovil loanee Ben Whitfield replied with a tame bouncing effort on 10 minutes prior to a glorious opportunity for Zoko to open the scoring.
The Ivorian forward raced by Valiants’ Nathan Smith, drew keeper Ryan Boot and sent a side-footer towards the bottom corner only for it to whistle just past the far post.
In the two meetings between the teams so far this season, Vale have looked dangerous in the air. Nathan Smith sent a free header over from six yards and Tom Pope followed up with lofty nod of his own. Warning shots.
The visitors grew into the first half and if not for some heroic defending would have tested Artur Krysiak.
Sowunmi and Connor Smith denied Whitfield with superb blocks and Bailey jumped in front of a powerful David Worrall attempt to keep Vale’s shots on target column clean.
As the rain continued to fall, the Stoke-on-Trent outfit continued to launch the ball up to Pope.
It almost proved effective on the half-hour mark with the big striker, a goalscorer against Town twice this term, connecting to Gavin Gunning’s whipped delivery but Krysiak was equal to it.
A long range Khan shot was all the next quarter of an hour had to offer but the game livened up in first-half stoppage time.
James played a beautifully crafted one-two with Surridge and was bundled over by Montano when he went to shoot. Referee Gavin Ward pointed to the spot and sent the Vale wideman off before Khan dispatched from 12 yards.
Vale were reduced to nine men early in the second half. No stranger to a dismissal at Huish Park, Gibbons received his marching orders for his second game running in Somerset for a lunge on Dickson.
Several visiting players took exception to Ward’s decision leading to the frustrated figures of Pope and Whitfield being booked. The duo were substituted shortly afterwards to avoid further punishment.
Yeovil struggled initially to make the most of their two-man advantage, although James gave it his best, collecting Khan’s cheeky, lobbed through ball but miscuing his eight-yard effort.
Then Gray was sent bursting down the wing and his fizzing delivery across the corridor of uncertainty looked destined to be tucked away by Surridge, who somehow missed the target.
James found himself in another forward position to warm Boot’s gloves before Khan fired over on the angle.
Town’s inability to capitalise on having more players on the field proved costly as Port Vale netted an equaliser late on to send the replay to extra-time.
Replacement Harness took a touch out of his feet on the edge of the box and curled a stunning strike past the despairing dive of Krysiak.
Three minutes into the first half of the further period, Khan put the Glovers back ahead.
Picking the ball up on the angle of the area, the former Barnsley trickster hit a stunning half-volley in off the woodwork to make it 2-1.
But another incredible strike levelled the tie up for Port Vale. Kay’s looped effort from distance hit the back of the net, sparking scenes amongst the away fans and players alike.
Their joy was short-lived as Zoko converted James’ low cross under 60 seconds later to put Town into the next round.
Yeovil Town: Krysiak, James, Sowunmi, N.Smith, Dickson, Gray (Green 77), Bailey, C.Smith (Worthington 27), Khan (Santos 115), Surridge (Olomola 69), Zoko.
Bookings: None.
Not used: Maddison, Davies, Mugabi.
Port Vale: Boot, Gibbons, Anderson, N.Smith, Gunning, Montano, Tonge (Reeves 93), Kay, Worrall (Harness 69), Whitfield (Denton 61), Pope (Barnett 61).
Bookings: Pope 54, Whitfield 58, Kay 109.
Sending off: Montano 45, Gibbons 54.
Not used: Lainton, Pugh, Turner.