Roberto Martinez's men had managed to find the net in every league fixture this term before this weekend, but endured a frustrating 90 minutes against the Spaniard's former club at Goodison Park.
Everton saw clear-cut goalscoring opportunities limited in the first half, despite dominating possession, and could count themselves lucky not to concede a penalty when Antolin Alcaraz appeared to handle the ball from a Jonjo Shelvey strike.
Swansea - who picked up their first-ever win over Everton in the League Cup earlier this season - looked brighter after the break, but were dealt a blow when Shelvey was given his marching orders for a second bookable offence 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 stay ninth.
Swansea returned to a more familiar starting XI following their 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.
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.
Swansea adopted a more defensive approach after going a man down and will be satisfied with a point under the circumstances, but Everton, who 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 successive league win.
