Premier League midfield duo Steven Pienaar of Everton and Fulham's Kagiso Dikgacoi were both selected.
Everton went close through a curling effort from Steven Pienaar, while Rui Patricio saved from Louis Saha.
Meanwhile, Everton's Steven Pienaar has joined up with his South African team-mates at their World Cup training camp.
But they responded well and Steven Pienaar converted Leon Osman's cross with a fine strike to open the scoring.
Marouane Fellaini headed over in a poor first half before Steven Pienaar's shot was well saved as Everton pressed.
Steven Pienaar also came close for the hosts when his 25-yard shot was palmed wide by keeper Julio Cesar.
Marouane Fellaini headed home the opener in the 42nd minute before Steven Pienaar doubled the lead in the 59th minute.
Marouane Fellaini gave Everton the lead when he slotted in Leighton Baines' cross before Tim Cahill headed in Steven Pienaar's corner.
Despite picking up three points at Ewood Park, Spurs suffered the last-minute loss of key midfielders Steven Pienaar and Luka Modric.
The defender received his first yellow for over-celebrating his goal and moments later was booked for a challenge on Steven Pienaar.
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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's Steven Pienaar hit the bar soon after while from Mikel Arteta's smart free-kick Jack Rodwell glanced a second-half header just inches wide.
Everton finally hit back in injury-time as striker Louis Saha, who replaced midfielder Leon Osman in the second half, netted after Steven Pienaar's shot was blocked.
Steven Pienaar raced clear and dinked the ball over Manuel Almunia to make it 2-1 but Rosicky's shot was deflected in to salvage a draw for Arsenal.
Midfielders Steven Pienaar (hamstring) and Leon Osman (stomach strain) have both missed recent games, while Anichebe was injured by a horror tackle from Kevin Nolan against Newcastle.
Everton manager David Moyes was forced to make a change on the hour when he introduced Diniyar Bilyaletdinov after Steven Pienaar was carried off on a stretcher.
Everton could have scored more, with both Johnson and Steven Pienaar hitting the post, as David Moyes' side secured their seventh consecutive, and their biggest, European win.
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What is undeniable is that the visitors did not look like a team without a league win, with Seamus Coleman and Steven Pienaar providing plenty of width.
The scare briefly roused Spurs and a curling Jermaine Jenas free-kick slammed against the upright, before substitute Steven Pienaar was denied by a brilliant close-range stop from Jaaskelainen.
Elsewhere, new South Africa captain Steven Pienaar , who replaced deposed Aaron Mokoena last week, is a big doubt for Saturday's critical Group G match at home to Egypt.
Joseph Yobo's 10th-minute header was followed by a close-range finish by summer signing Distin from another corner seven minutes later, and a superb right-foot effort by Steven Pienaar on 37.
Steven Pienaar 16th minute goal sees David Moyes' side move to 59 points and within three points of Spurs, although the Merseyside club have played one more game than the North Londoners.
CNN: Spurs drop points at Wigan to dent Champions League bid
The Milan skipper, making his 450th appearance for the Rossoneri, confronted Spurs striker Peter Crouch before collecting a booking that will rule him out of the second leg for a lunge at Steven Pienaar.
Everton were struggling to exert any authority in midfield, and Steven Pienaar was booked after a succession of fouls - indeed he was lucky not to be shown the red card for persistent fouling.
Refusing to discuss the club's move for Steven Pienaar, which has stalled due to the Everton midfielder's wage demands, Ancelotti insisted he had never felt his job had been on the line in recent weeks.
The striker, who was surprisingly brought on for South Africa's best player, midfielder Steven Pienaar, controlled a left-wing cross by Siphiwe Tshabalala with his knee and stabbed the ball past Iker Casillas in the 73rd minute.
It was Everton's two late substitutes who clinched the three points as Dan Gosling slid in Steven Pienaar's cross with 14 minutes left and Jack Rodwell made it three with a solo effort in the dying seconds.
Twice Jo was sent clear on goal, first by Tim Cahill and then by Steven Pienaar, but first the Brazilian shot straight at keeper Petr Cech and then he slipped at the vital moment, fluffing his effort wide of goal.
Largely dictating the play despite that early setback, the visitors nearly scored in the 17th minute when Baines jabbed a nicely-weighted pass inside for Steven Pienaar who curved the ball casually out of Brad Freidel's reach but against the bar.
The nimble and tricky Osman had replaced the injured Steven Pienaar on the left of the Toffees midfield and it was his cross from which Johnson prodded into his own net to give the visitors a deserved lead as they notched their first away goal in the Premier League this season.
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