Phil Jagielka, Bryan Oviedo and Ross Barkley were all denied by Sunderland goalkeeper Vito Mannone.
But the Black Cats, three points from safety, produced a spirited display.
With fellow strugglers Fulham and Crystal Palace also winning away, Sunderland
remain bottom of the table
and approach the midway point of the season knowing they still have much work to do if they are to stay up.
But victory over one of the Premier League's most in-form sides will provide Gus Poyet's side with a major boost.
Sunderland had failed to score in four of their previous six league games.
Yet the Capital One Cup semi-finalists were gifted their first goal on the top-flight stage for four hours and 18 minutes by midfielder Leon Osman, recalled to the side for his 302nd Premier League Toffees outing.
Rare away joy for Sunderland
Sunderland's previous away win in the Premier League came on 14 April at Newcastle. Since then the Black Cats had failed to win in 10 attempts on the road before this victory.
The 32-year-old's poor back pass let in Ki, the South Korean falling under a Howard challenge as he attempted to round the Everton keeper.
Referee Lee Probert pointed to the penalty spot, although replays were inconclusive about the amount of contact from Howard.
on loan from Swansea,
picked himself up to send the ball beyond substitute keeper Joel Robles, leaving Everton, down to 10 men and unbeaten in the Premier League at Goodison since 30 December 2012, shaken.
Robles, making his first Toffees league appearance since
signing from Atletico Madrid in July,
denied Sebastian Larsson from doubling the Black Cats lead with his right palm before the hosts produced a second-half rally.
Everton defeat not a set-back - manager Roberto Martinez
Sunderland keeper Mannone did well to keep out Jagielka's header following a corner, while the Italian produced an even better stop to deny Oviedo.
After Steven Fletcher had spurned a fine chance to put the game out of Everton's reach, Barkley was foiled by Mannone.
Everton's frustration was complete when Sunderland forward Fabio Borini,
on loan from Liverpool,
cleared a header from substitute Nikica Jelavic off the line.
Everton manager Roberto Martinez:
"Goalkeeper Vito Mannone was man-of-the-match for Sunderland but we should have scored.
"Obviously we were a bit sloppy in that situation [the red card].
"The fans will be disappointed with the score but everyone saw the incredible performance we had after the interval.
"I'm a lot happier today than other days when we have won because we learned a lot about the squad."
Sunderland manager Gus Poyet:
"Even with 10 men they [Everton] were the best team we have played against.
"It was very difficult and it's incredible that we have beaten them.
"The players are tired, they gave everything possible for the fans."
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