Michael Laudrup's men followed up their Capital One Cup victory over Chelsea by defending resolutely as they took a point away from Merseyside on Saturday afternoon
Ashley Williams and Chico Flores produced a defensive masterclass for the Swans, who tired toward the final moments of what was an entertaining encounter at Goodison Park.
Everton made just one change following their FA Cup win over Cheltenham, with Steven Pienaar returning on the left in place of Bryan Oviedo. Michael Laudrup, meanwhile, named Gerhard Tremmel, Wayne Routledge, Leon Britton and Jonathan de Guzman on the bench following their exploits in the midweek win over Chelsea, with Michel Vorm, Kemy Agustien, Dwight Tiendalli and Nathan Dyer their replacements.
The Toffees started brightly, with Marouane Fellaini stinging Vorm's palms after some delightful link-up with Pienaar on the left-hand side. However, it did not take long for the Swans to evidence their well-documented threat on the counterattack, with their best chance coming moments after a dangerous Everton move broke down.
Dwight Tiendalli's clearance - one which was arguably unnecessary given the ball appeared to be going wide – landed to Hernandez, who performed an impressive flick to escape the attentions of Phil Jagielka. Unfortunately for the visitors though, the Spaniard could not find Dyer's run up the right-hand side after cutting inside.
With 30 minutes gone, David Moyes' side went even closer to breaking the deadlock as Fellaini's knock down to Victor Anichebe was kept out by Vorm before being hacked away desperately by Ashley Williams. Yet, in keeping with the theme of the game, the Swans went close next as Michu's lofted effort was flicked onto the bar by Tim Howard.
There was, of course, time for one more chance in the half, as Seamus Coleman wriggled beyond two opposition defenders before crossing for Fellaini, but the Belgian's shot was weak and easily held by Vorm.
The second period started in equally blistering fashion, as Phil Neville collected the ball from Coleman before forcing Vorm into a save from close range. And just moments later Sylvain Distin headed over from a corner to underline the Toffees' status as the dominant force upon the resumption of play.
Swansea began to move deeper defensively as Everton dominated territorially, forcing Laudrup to look to Routledge and De Guzman off the bench as he searched to put the Swans’ stamp back on the game.
However, the hosts continued to threaten, with Jagielka somehow heading over from six yards out after another brilliantly delivered Leighton Baines corner.
A raft of substitutions late on offered hope of a breakthrough from either side, though none was forthcoming as the game lost its fluidity.