Sunderland earn replay with Everton
Bardsley fired the Black Cats into an early lead with a superb strike but Everton had the better of the chances throughout and claimed a deserved 24th-minute equaliser through Cahill.
Royston Drenthe, who had a strong penalty appeal turned down early on, hit the woodwork for Everton while Simon Mignolet made a fine double save to deny John Heitinga and Nikica Jelavic in the closing minutes.
Because of their dreadful recent record against Everton, it was a result Sunderland, without suspended duo Lee Cattermole and Stephane Sessegnon, might have settled for beforehand. The Black Cats have now not beaten the Merseysiders in 16 games stretching back to 2001.
The cup competition provides a fine opportunity for both clubs this season and that made for a frenetic end-to-end start, although the game slowed and became scrappy in the second period.
Sunderland had an early chance when James McClean glanced a header wide from a Sebastian Larsson free-kick, while Everton were also into their stride quickly as Leon Osman headed over from Seamus Coleman's cross.
Everton manager David Moyes was criticised for making six changes for the midweek derby loss at Liverpool but his intentions were clear as he reverted to side which beat Tottenham Hotspur last week. His players were keen to attack and Drenthe appealed for a penalty after going down under a challenge from Craig Gardner in the eighth minute.
Bardsley breaks deadlock
Replays suggested the Dutchman may have had a case but referee Andre Marriner was not interested. Sunderland grabbed the lead in the 12th minute with a ferocious low strike from Bardsley from 30 yards.
Everton may have been expecting a cross as Jack Colback prepared to take a free-kick on the right but he played short to Bardsley and the defender fired through a crowded penalty area. The hosts tried to produce a quick response but Osman and Marouane Fellaini both missed the target from distance and Heitinga had a free-kick deflected wide.
Everton got their threatened equaliser as Baines delivered a typically dangerous cross from the left. Jelavic misdirected his header but Cahill followed up with one of his own to beat Mignolet in trademark fashion, although it was only his second goal of the season.
Jelavic threatened another as he raced onto a long ball from deep but could only screw a shot wide under pressure from Wayne Bridge. Sylvain Distin had a better chance after Cahill tested Mignolet with another header which the goalkeeper could only punch off the line.
The ball fell invitingly for Distin but Sunderland escaped as the defender blasted wide from close range. Drenthe was unlucky just before the break as he curled a free-kick against the bar from 30 yards after Coleman was fouled by Bridge. Osman tried another shot from just outside the area but Mignolet gathered easily.
Mignolet on form
Everton continued to take the attack to the visitors in the second half, with Cahill bursting forward and trying his luck from long range. Mignolet got down but spilled the ball and it needed Bridge to clear the danger. Sunderland's Nicklas Bendtner found the net after being played through by McClean but the flag had long been raised for offside.
Coleman cut inside two defenders on another Everton raid but fired his left-footed shot into the side-netting. Cahill had another chance as Fellaini headed down an Osman corner but his shot on the turn lacked the power to beat Mignolet.
Jelavic won a header as Baines whipped in a free-kick but his effort flew over and the Croatian had a penalty shout turned down after heading another against John O'Shea's arm. Jelavic suffered more frustration as he headed straight at Mignolet and then had a shot blocked.
Sunderland, perhaps thinking in terms of a replay, attempted to take the edge out of the game as time ran down but Everton finished strongly. Mignolet produced a fine double save to prevent the Toffees snatching a winner four minutes from time.
Osman threatened to shoot but instead clipped a cross to Heitinga, whose header had Mignolet at full stretch. Jelavic pounced on the rebound but Mignolet recovered to block as Sunderland held on.