It was the South African who fashioned the game's first real opening, nine minutes in.
His clipped, in-swinging cross from the left was inviting, and Nikica Jelavic did well to evade Ben Davies to reach it at the far post.
The Croatian's contact, though, was not clean, and the ball drifted agonisingly wide of Michel Vorm's right-hand upright.
Jelavic went close again soon after, meeting Baines' corner from the right on the run, but seeing his glancing header kicked away from the post by Tiendalli.
Anichebe was next to find himself with a sight of goal, after Fellaini had climbed well above Chico to nod Jagielka's diagonal ball across goal.
Anichebe's close-range, stooping header was half-stopped by Vorm, before Williams completed the clearance from the goal-line.
Swansea, by contrast, were offering up precious little. Laudrup was apoplectic as Nathan Dyer wasted one promising break with a flashy backheel, whilst Hernandez was particularly careless with his final pass.
It was 36 minutes before the 16-goal Michu was given a sight of goal.
The Spaniard was sent scampering through by Dyer's pass but, with the angle prohibitive, Howard was able to get a hand on his attempted chip, turning the ball onto the frame of the goal. It was to prove Swansea’s last real chance of the afternoon.
Everton, meanwhile, continued to probe. Fellaini shot weakly at Vorm after enterprising work from Seamus Coleman, while Osman, eight yards out, failed to make a good connection with another Baines set-piece as Jelavic jumped in front of him.
The theme continued after the break. Vorm denied Phil Neville at the front post early in the half, while Distin headed over after beating Chico to Baines' corner on the hour mark. Hernandez's meek 20-yarder, which Howard fielded comfortably, was about as adventurous as Swansea got.
There were half-hearted penalty appeals when Jelavic tumbled under Williams' challenge at the other end, and groans as Jagielka sent a free header over from yet another Baines free kick.
The miss summed up Everton's frustration, as the Goodison crowd was forced to endure a final 10 minutes which saw both Vorm and Tiendalli booked for time-wasting.
The result leaves Everton on the cusp of the Champions League qualification spots, three points behind fourth-placed Tottenham.
It seems churlish to be critical in a season which has seen so much progress at Goodison – and on a run which has seen Everton lose just three of their last 31 league games – but the fact that this was the Toffees' 10th draw of the campaign is not one to be ignored.
"It could quite easily cost us if we don't turn these into wins,” admitted Moyes afterwards.
His side can play better than this, but if they heed his advice, they will not be far away from the top four come the