Mikel Arteta's early free-kick and two second-half strikes from Tim Cahill put the Toffees in command.
Tim Cahill secured the win with a stooping close range header from a Leighton Baines cross in injury-time.
As the rain began to lash down the game turned scrappy with Tim Cahill and Sturridge off-target with half-chances.
Red Bulls teammate Tim Cahill says the issue is "not about points" but rather about disciplining the fans involved.
Tim Cahill headed home the opener after four minutes before setting up Leighton Baines to curl in Everton's second.
Everton finally got out of their own half when Tim Cahill broke but his ball into the box was misdirected.
Tim Cahill scored a trademark header after a neat build-up between Mikel Arteta and Leighton Baines, who provided the cross.
Kewell's Galatasaray team-mate Lucas Neill, Everton midfielder Tim Cahill, and Fulham goalkeeper Mark Schwarzer were also included in Verbeek's squad.
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Marouane Fellaini gave Everton the lead when he slotted in Leighton Baines' cross before Tim Cahill headed in Steven Pienaar's corner.
The Toffees, without their top scorer Tim Cahill, relied on Saha, Rodwell, Marouane Fellaini and Diniyar Bilyaletdinov to provide the bullets.
Tim Cahill's early overhead strike put his side in front, and Bolton's task was made more difficult by Bruno N'Gotty's 45th-minute sending-off.
He had played a starring role, Marouane Fellaini had been outstanding and Tim Cahill had put in his best performance in weeks.
Distin inadvertently diverted a low cross to Marney and, symbolic of Everton's fortunes, the Hull midfielder's shot went in off Tim Cahill.
Steven Pienaar marked his 100th game for the club with a fine left-footed finish after a clever backheel from the inspirational Tim Cahill.
Everton midfielder Tim Cahill played under Hiddink for the Australia national side and said he retained a lot of respect for the Dutchman.
Louis Saha missed a gilt-edged chance when Wigan keeper Ali Al Habsi blocked his strike before Tim Cahill's header hit the Latics post.
Despite playing without creative midfielder Tim Cahill, an Australian international call-up, the Red Bulls generated plenty of scoring chances in the scoreless first half.
Fellaini had the first chance after a delightful first-time pass from Tim Cahill, but the big-haired Belgian's poked effort was brilliantly blocked by Ruddy.
Everton boss David Moyes welcomed back Tim Cahill after a two-month injury lay-off but the Toffees sorely missed the suspended Mikel Arteta's creativity in midfield.
Tim Cahill, who scored a late game-winning goal against on May 4, sent a long pass from midfield that George went up to head and missed.
In other MLS news Tim Cahill joins the New York Red Bulls while Honduran midfielder Mario Martinez moves to Seattle Sounders on loan from Real Espana.
But it was Everton who dominated in terms of chances, with Joleon Lescott hitting a post and Louis Saha and Tim Cahill wasteful in front of goal.
Everton were boosted by the return of Tim Cahill after suspension and Leon Osman after injury - and David Moyes's side took charge in the first 45 minutes.
Less than 60 seconds later, however, Everton went up field, Tim Cahill's neat back-flick found Yakubu and the former Middlesbrough striker's curling shot beat the hapless Hakon Opdal.
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Dzeko fired narrowly wide for City as they attempted to put daylight between themselves and an increasingly confident Everton, who made a change by replacing Rodwell with Tim Cahill.
"I'm not going to talk about the sending-off, I'll only talk about what happened prior to it, " added the Everton boss, whose team's solitary goal came from Tim Cahill.
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It was Pienaar who produced the game's first meaningful shot after Tim Cahill chested the ball down for the South Africa international, but Schwarzer safely gathered low to his left.
The hosts went ahead with a thumping drive from Dmitri Likhtarovich but Marouane Fellaini drew Everton level from a Leighton Baines free-kick and Tim Cahill won it with 13 minutes remaining.
Sasa Ognenovski made sure Australia went in with a 2-0 cushion at half time as he fired home a loose ball after Tim Cahill had nodded down a David Carney center.
Australia trailed Oman, who have won just once in six qualifiers, 2-0 before New York Red Bulls midfielder Tim Cahill halved the deficit when heading home from a Luke Wilkshere corner.
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