Neville had watched the last two matches as an unused substitute but was the unlikely match-winner when recalled to the side by David Moyes for this tie.
Various attacking combinations had failed to find a breakthrough after Chris Brunt's penalty midway through the second half had given the visitors the lead.
But Marouane Fellaini gave Everton a lifeline they barely deserved with an unstoppable 89th-minute volley.
And Neville, who has fallen behind Tony Hibbert in the pecking order for right-back, showed his forwards how it should be done with a quality left-footed finish from outside the area in the first half of extra-time.
For long periods, however, it appeared the Toffees would struggle to score one goal never mind two.
Just as they had done against Wigan on Saturday - when they came from behind to win - Everton got an early grip of the game but struggled to make it count.
Summer loan signing Denis Stracqualursi, on his first start, fired an early shot wide with Fellaini flicking a header past the far post from Leighton Baines' left-wing cross, Jack Rodwell bundling a shot well off target and Royston Drenthe, also on full debut, forcing Marton Fulop into a save at his near post.
The Baggies were similarly toothless as only Peter Odemwingie looked threatening.
He left Phil Jagielka struggling to keep his footing as he danced past him and centre-back partner John Heitinga on the edge of the area only to shoot wide.
Odemwingie produced a similar result with another long-range shot having run through the inside-right channel.
The second half began the same as the first but Seamus Coleman should have done better having worked an opening for himself on the right of the area, shooting tamely at Fulop.
Another Drenthe free-kick picked out Fellaini at the far post and he nodded back to Ross Barkley whose wild swing diverted the ball to Stracqualursi but he hooked a shot straight at the goalkeeper.
But as had been the case at the weekend, the visitors hit Everton with a counter-attack goal in the 56th minute.
Somen Tchoyi's through-ball sent Odemwingie clean through and as Jan Mucha raced off his line there was going to be only one winner.
The goalkeeper escaped with a yellow card for his foul and Brunt smashed home the penalty.
Everton ramped up the pressure in the closing stages with Apostolos Vellios, who had replaced Barkley just past the hour, glancing a header wide, Fellaini nodding weakly into Fulop's arms and Baines drilling a 20-yard free-kick into the wall.
With a minute of normal time remaining Fellaini's well-controlled volley from the edge of the area found the bottom corner of the net to force an extra 30 minutes.
Odemwingie could have won it in the fifth minute of added time after weaving past Jagielka and Mucha but his angled shot flew over.
Neville did not make the same mistake in the 103rd minute when Drenthe cut the ball back to him and he curled a left-footed shot past Fulop from 25 yards.