Ross Barkley's fine free kick ensured Everton would sit in the top five of the Premier League at Christmas as they won at Swansea. A Bryan Oviedo own goal had earlier cancelled out a sensational strike from Seamus Coleman all in the second half.
Similar styles and systems meant this match was decidedly tactical from the outset. That naturally diminished the amount of goalmouth action. Gerhard Tremmel in the Swans goal saved routinely from Kevin Mirallas, but that chance apart defences were on top.
Barkley fell over at the crucial moment when in behind the home side after a goalless first half. Seamus Coleman then went on a tear, but last-ditch defending denied Everton again. Swansea pair Chico Flores and Ashley Williams earlier put in several blocks.
Gareth Barry blazed over after an hour with the latest in a glut of Toffees chances. Barkley then went close as he drove at the heart of Laudrup's lads, forcing Tremmel to tip his effort onto the top of the bar.
Everton finally broke the deadlock with 25 minutes left through Coleman as he bent a rasping strike inside the near post. The Swans quickly levelled, however, as Dwight Tiendalli's shot deflected in off Oviedo.
Seven minutes from time the Toffees won it with Barkley firing in a free kick off the underside of the bar. Ben Davies's naïve foul on James McCarthy cost Swansea a point.
Tim Howard also had to be on hand to parry away Pablo Hernandez and Roland Lamah efforts in the closing stages, so the Swans came up short.
Roberto Martinez finally has a league win over his former club and Everton know they will be no lower than fifth in the Premier League at Christmas. Swansea meanwhile remain top of the bottom half but their wait to make history and beat the Toffees goes on.