MUCH-IMPROVED Albion's Carling Cup adventure was ended at the third-round stage by Phil Neville's extra-time stunner.
The Everton skipper curled home a superb 18-yard effort in the 102nd minute, giving keeper Marton Fulop no chance.
It was tough on Albion, who shaded the match in normal time and deservedly took the lead through skipper Chris Brunt's 57th-minute penalty.
But there was late agony for the Baggies, after Everton took the tie into extra-time through Marouane Fellaini's fine 89th-minute leveller.
Hodgson made nine changes from the dismal defeat at Swansea on Saturday, with Jonas Olsson and Peter Odemwingie the only survivors.
Brunt was recalled to the starting XI while Academy graduate Saido Berahino was named among the subs - his first appearance in a senior matchday squad.
Changes were the order of the evening, with David Moyes making six from Everton's home win over Wigan.
Albion started in a new-look 4-3-3 formation and one noticeable improvement from their Liberty Stadium display was their pressing of the opposition when out of possession.
It wasn't until the eighth minute that either side mustered an effort at goal, with Somen Tchoyi jinking inside Neville before firing well wide from 30 yards.
On 14, Everton keeper Jan Mucha was a little red-faced when he allowed a John Heitinga back-pass to slip under his foot and creep goalwards. Thankfully for the Toffees No.1, there wasn't much pace on the ball and he was able to retrieve it inside his six-yard box.
Two minutes later, Albion probably should have been ahead.
Tchoyi latched onto George Thorne's through-ball and after Phil Jagielka and John Heitinga got in each other's way as they tried to dispossess the Albion man, he muscled his way through, only to shoot wide from just inside the box.
Everton recorded their first shot of the game on 18, but Royston Drenthe failed to trouble Fulop's goal with an ambitious 30-yard free-kick.
The Toffees were enjoying the lion's share of possession but it was Albion who continued to look more threatening.
On 26, a woeful pass by the lively Ross Barkley gave Odemwingie a chance to run at the home defence and he skipped inside Heitinga before shooting wide of the right-hand post from 15 yards.
Two minutes later, Everton finally carved out a decent chance and Fellaini will feel he should have done much better when he flashed a ten-yard header wide from Leighton Baines' inch-perfect centre.
After Albion only half-cleared a Drenthe free-kick on 33, Jack Rodwell saw a 25-yard snapshot fly harmlessly wide of Fulop's far post.
The Baggies kept Everton pegged back during the closing stages of the half but, on several occasions, their final pass was found wanting.
The second half suddenly burst into life in the 49th minute when Seamus Coleman stormed past three Albion defenders before sending an acute drive straight at Fulop from 15 yards.
The Merseysiders went even closer a minute later when Fellaini rose at the back post to head Baines' diagonal cross back into the danger-zone. Fulop did well to keep out Denis Stracqualursi's snapshot with a reflex save before Albion cleared their lines.
Albion responded with a period of pressure of their own and from a second successive corner, Thorne was inches away from sliding home Brunt's centre at the back post.
The Baggies were turning the screw and snatched a lead they just about deserved on 57.
They were awarded a penalty when Odemwingie was pulled down by Mucha as he just beat the Everton keeper to Tchoyi's through-ball.
After Odemwingie missed from the spot at Norwich two matches ago, Brunt took over the penalty duties and he gave Mucha no chance with a powerful low effort.
Albion broke dangerously again four minutes later and Neville did well to get in the way of Tchoyi's ferocious 20-yard shot which looked goalbound.
The visitors continued to look the better side and Baines had to produce an acrobatic interception to prevent Simon Cox stabbing home Dorrans' low centre.
Everton failed to clear their lines and after Brunt had directed the ball back towards the six-yard box, Cox saw a back-header loop straight to Mucha.
Moments later, Dorrans fizzed a vicious 25-yard effort just over.
Everton did not threaten again until the 79th minute when sub Apostolos Vellios glanced a header just wide of the far post from Coleman's centre.
The Toffees went close again two minutes later when Fellaini headed straight at Fulop after Baines had ghosted past Cox and centred.
It was a case of déjà-vu for Everton in the 86th minute, when Fellaini again nodded directly at Fulop, this time after Brunt had headed Neville's right-wing cross straight up in the air.
At the other end, Mucha did brilliantly to push away Brunt's in-swinging free-kick as sub James Morrison slid in.
Albion had defended superbly all game - only to concede in the 89th minute.
Fellaini was first to seize on a loose ball on the edge of the Baggies box before swivelling and directing a superb half-volley inside the far post.
Rodwell and Odemwingie both had chances in stoppage time but neither could convert and the game crept into extra time.
Tchoyi's brute strength created the first chance after the restart when he outmuscled Heitinga in the box near the byline.
The Cameroon forward dragged the ball past the on-rushing Mucha, only to lift his shot over the bar from six yards.
But it was Everton who scored next - with another well-taken goal.
On 102, Drenthe nudged the ball back to Neville who curled an unstoppable first-time effort inside the far corner from 18 yards.
It was Everton who threatened first in the final period, with Rodwell hacking a half-volley well over from Baines' corner.
Three minutes from the end, Everton sub Magaye Gueye saw a deflected shot whistle just wide of the near post.
Albion created one last chance when Dorrans fed sub Paul Scharner in the box, but the Austrian's near-post drive was pushed wide by Mucha when he may have been better served crossing to Odemwingie.
EVERTON (4-5-1): Mucha; Neville, Heitinga, Jagielka, Baines; Coleman (Gueye 79), Rodwell, Fellaini, Barkley (Vellios 62), Drenthe; Stracquallisi (Cahill 62). Subs not used: Howard (gk), Hibbert, Bilyaletdinov, Distin.
ALBION (4-3-3): Fulop; Jones, McAuley, Olsson (Dawson 103), Mattock; Dorrans, Thorne (Scharner 96), Brunt; Odemwingie, Tchoyi, Cox (Morrison 83). Subs not used: Daniels (gk), Jara Reyes, Mulumbu, Berahino.
GOALS: EVERTON - Fellaini (89), Neville (102). ALBION - Brunt (pen 57).
BOOKINGS: EVERTON - Mucha (foul 56), Drenthe (foul 63). ALBION - Dorrans (persistent fouling 33).
REFEREE: J Moss (Yorkshire).
ATTENDANCE: 17,647.
ALBION STARMAN: Peter Odemwingie.
PIC: SPOT ON...Brunt smashes home Albion's 57th-minute penalty.
