The visitors were three ahead at the break through Denilson - from the edge of the box - and two headed goals from Thomas Vermaelen and William Gallas.
Things got worse for the Toffees in the second period when Cesc Fabregas grabbed two for himself to underline an outstanding personal performance.
Eduardo netted the sixth before Louis Saha scored a consolation goal in injury-time for Everton.
Elsewhere Didier Drogba netted a fortunate winner in stoppage time to give Chelsea a 2-1 victory over Hull City in Carlo Ancelotti's first Premier League game in charge.
Hull's Republic of Ireland midfielder Stephen Hunt - signed only on Thursday from Reading - fired the visitors into a shock lead on 27 minutes.
However, the title favourites were level 10 minutes later when Drogba crashed in a dipping 25-yard free-kick.
Then, with two of the six added minutes played, just as it looked like the Tigers had secured a point following a stern defensive second-half display, Drogba's chip into the area from a tight angle looped over keeper Boaz Myhill and into the back of the net.
Manchester City started to see a return for their summer spending spree with a goal after just three minutes from £25million striker Emmanuel Adebayor in a 2-0 win at Blackburn.
The Togo international, signed from Arsenal, converted Shaun Wright-Phillips' cross to open his account.
City boss Mark Hughes, who has laid out almost £100million, also handed competitive debuts to defender Kolo Toure and England midfielder Gareth Barry - but former Manchester United striker Carlos Tevez had to be content with a place on the bench.
Midfielder Stephen Ireland wrapped things up with a second goal for the visitors in stoppage time.
City's victory was one of five away wins out of the first seven Premier League games.
New Wigan manager Roberto Martinez saw his side produce an unexpected 2-0 victory at much-fancied Aston Villa.
Colombia forward Hugo Rodallega (31) drilled in a 25-yard effort to put the visitors ahead at half-time, with Jason Koumas (55) securing the three points with a close-range finish following a fine run by former Newcastle midfielder Charles N'Zogbia.
Last season's Coca-Cola Championship winners Wolves went down 2-0 at home to West Ham.
Mark Noble (22) put the Irons ahead midway through the first half, and following a string of good saves by England goalkeeper Robert Green as the hosts pressed for an equaliser, defender Matthew Upson (69) headed in a second from a corner to put the game beyond Wolves.
Darren Bent marked his debut for Sunderland with a fifth-minute winner against Bolton at the Reebok Stadium.
The £10million signing from Tottenham headed in from close range following a left-wing cross by Steed Malbranque to give new Black Cats boss Steve Bruce a winning start as they held off a late onslaught by the home side.
Fulham have already been in Europa League qualifying action and continued their early-season momentum as they beat Portsmouth 1-0 at Fratton Park.
Clint Dempsey's shot deflected off striker Bobby Zamora (13) and past England keeper David James, which was enough to collect all three points.
Play-off winners Burnley were handed a tough welcome back to the top flight as Stoke won 2-0 at Britannia Stadium.
Ryan Shawcross (19) nodded in a free-kick from Liam Lawrence, before an own goal by Stephen Jordan (33) doubled the home side's advantage and there was no way back for the Clarets.