The Toffees were shut out for the first time in the Premier League this season.
Everton failed to score for the first time this season as it was held to a scoreless Premier League draw by Swansea City on Saturday.
Roberto Martinez's men had managed to find the net in every league fixture this season before this weekend, but endured a frustrating 90 minutes against the Spaniard's former club at Goodison Park.
Everton saw clear-cut scoring opportunities limited in the first half, despite dominating possession, and could count itself lucky not to concede a penalty when Antolin Alcaraz appeared to handle the ball from a Jonjo Shelvey strike.
Swansea — which picked up its first-ever win over Everton in the League Cup earlier this season — looked brighter after the break, but was dealt a blow when Shelvey was given his marching orders for a second bookable offense with 18 minutes to go.
However, Garry Monk's men held firm to cling on to a hard-earned point that sees them remain sixth, while Everton stays ninth.
Swansea returned to a more familiar starting XI following its League Cup elimination at the hands of Liverpool on Tuesday, while Martinez could only include Romelu Lukaku among his substitutes after the Belgian injured a toe in last weekend's victory over Burnley.
Samuel Eto'o led the Everton line after netting a brace in that 3-1 win at Turf Moor, and could have put the home side in front with a half-volley inside the opening minute — Lukasz Fabianski making the save.
Everton had the better of the early chances and Leighton Baines curled a free kick just wide of the target from 20 yards after Eto'o had been felled by Ki Sung-yueng.
Swansea boss Monk has had a number of run-ins with officials in recent weeks and the visitors were appealing for a penalty in the 20th minute when a Shelvey strike appeared to hit the arm of Alcaraz, but referee Kevin Friend opted against pointing to the spot. Replays suggested Shelvey had every reason to feel aggrieved.
Alcaraz had to be replaced by Muhamed Besic after coming off worse in a collision with Wilfried Bony — Gareth Barry switching to a central defensive role as a result — but it was Everton that ended the half in the ascendancy.
Barry's inexperience in the back four showed when he was almost caught in possession by Wayne Routledge shortly after the restart, before Shelvey fired wide having been teed up by Bony.
Everton struggled to recapture its first-half dominance after the break, but finally gave the visiting defense something to think about when Aiden McGeady curled just wide after creating a shooting angle just outside the box.
Shelvey, who has already served a one-match suspension for accumulating yellow cards this season, was made to pay for kicking the ball away in the first half when he received a second booking for cynically blocking James McCarthy in the 72nd minute.
Swansea adopted a more defensive approach after going a man down and will be satisfied with a point under the circumstances, but Everton, which saw Lukaku just fail to get on the end of a fizzed cross in the final seconds, may view the match as a missed opportunity to record a third straight league win.