Everton extended their unbeaten league record against Swansea but failed to find the killer touch to take all three points at the Liberty Stadium.
The Toffees carried the greatest threat for much of an entertaining contest which somehow remained goalless despite the amount of clear-cut chances created.
But they were probably relieved to take a point when substitute Kevin Mirallas, who had only been on the pitch for 112 seconds, received a straight red card for a dangerous tackle on Modou Barrow.
Ross Barkley shone throughout but Everton could not take advantage of his excellent approach play, with Romelu Lukaku the biggest culprit in front of goal for the visitors.
Swansea also had their moments to bounce back from last weekend's shock defeat at Watford with Tim Howard saving smartly on a few occasions - but Everton might have had strong claims for a penalty when Ashley Williams appeared to trip Steven Naismith late on.
Everton's proud record of never having lost to Swansea in the league stood at 20 games but the Toffees suffered a blow before kick-off when Seamus Coleman failed to recover from a hamstring injury.
Tyias Browning was handed his first Premier League start at right-back and with Brendan Galloway, who was only making his seventh top-flight appearance, on the left there was an inexperienced look about Everton's full-backs.
It was an area Swansea were no doubt keen to exploit, particularly as Jefferson Montero had returned to the left side of their attack following a thigh problem, and Browning was given an indication of the task ahead of him inside five seconds when the Ecuadorian flier went past him.
Everton were desperate to protect Browning whenever they could, often ending up with three or four players around Montero, but it was the visitors who produced the first moments of real danger.
Williams got in the way of a Lukaku free-kick flying towards the Swansea goal and Phil Jagielka's headed over the resulting corner from Barkley.
Swansea's early momentum had been slowed by a head injury to Andre Ayew, the Ghanaian able to continue after treatment, but Bafetimbi Gomis went close twice in the space of 60 seconds.
A frantic first half came to a stop when Barry blocked Gomis' shot and the second period started in similar fashion when Barkley beat Ayew and Arouna Kone failed to prod home his inviting cross.
More teasing approach work from Barkley almost ended with Galloway's first Everton goal after he had turned smartly inside the area, while Lukaku failed to reward the visitors' growing dominance by firing wildly over with only Fabianski to beat.
Gylfi Sigurdsson sent a Swansea free-kick into the side netting but Everton almost took the lead when Barkley's deflected shot fell to Lukaku and Williams covered his goalkeeper well to hack off the line.
But Howard was also needed at the other end as he batted away firm drives from Montero and Gomis to ensure the Everton defence remains unbreached away from home in the league this season.