Shakhtar Donetsk, who won the UEFA Cup in its last year, Sporting Lisbon and FC Salzburg all won to maintain their 100 percent records at the halfway mark in the group stage.
AS Roma ensured an unhappy night for English teams with an injury-time equaliser in a 1-1 draw at Fulham, while city rivals Lazio also struck in added time to beat Villarreal 2-1.
The top two in the 12 groups qualify for the knockout stage where they will be joined by 10 teams eliminated from the Champions League as third-place finishers.
Everton travelled to Lisbon with six points from their first two Group I outings but were plagued by injuries and fell behind to a goal from Argentine Saviola after 14 minutes.
The visitors were taken apart after the break when Paraguay striker Oscar Cardozo scored twice in a minute, turning in crosses from Saviola and Angel Di Maria, and Brazil defender Luisao headed in the fourth goal from a corner in the 52nd.
Saviola added a fifth late on to hand Everton their heaviest defeat in Europe after a result that left Benfica level with them at the top of the group on six points from three games.
EARLY LEAD
Brazil's Fernandinho gave Shakhtar Donetsk an early lead from the penalty spot against Toulouse before his compatriot Luiz Adriano added two more.
Czech Tomas Huebschman also scored to round off a 4-0 win for the Ukrainians who top Group J with nine points.
Norwegian Brede Hangeland headed Fulham in front after 24 minutes of their Group E game at Craven Cottage.
Roma's Jeremy Menez had a 78th minute penalty saved by Mark Schwarzer but Marco Andreolli rifled home a volley in the third minute of stoppage time to level the scores at 1-1.
Fulham fullback Stephen Kelly was sent off for the incident leading to the penalty.
Basel overtook Fulham at the top of the group, winning 2-0 at CSKA Sofia with two goals from Switzerland's record international scorer Alex Frei.
Sporting Lisbon maintained their perfect record after three games in Group D with only minutes to spare as Joao Moutinho gave them a 2-1 win at Latvia's Ventspils.
Lazio also pulled off a dramatic 2-1 victory despite playing the last quarter of their Group G game with 10 men after Brazilian midfielder Matuzalem was sent off.
Substitute Tommaso Rocchi's last-gasp winner left Lazio with six points, three behind Salzburg who beat Levski Sofia 1-0 with a Dusan Svento goal in first-half injury time.
WERDER HELD
Werder Bremen, beaten by Shakhtar in the UEFA Cup final in May before that competition was replaced by the Europa League, looked set to maintain their 100 percent record in Group L.
Peru striker Claudio Pizarro put them 2-0 ahead at Austria Vienna but Emin Sulimani pulled one back and Brazilian striker Schumacher struck an 87th minute equaliser for the hosts.
The 2-2 draw means
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