A tale of two goalkeepers renewed hope for Sunderland. Tim Howard was sent off with 70 minutes to play and Everton's unbeaten home record in the Premier League in 2013 disappeared with him. Vito Mannone excelled throughout and Gus Poyet received "the best present you could ever give me", with his keeper's defiance typifying Sunderland's refusal to be cut adrift.

Roberto Martínez's 10 men responded valiantly to the calamitous mistake that resulted in Howard's dismissal and Ki Sung-yueng's match-winning penalty but could find no way past Mannone. "Everton are the best team we have played against this season, even with 10 men," said Poyet as he savoured the first away win of his Sunderland reign.

The Everton manager was more pointed in his praise. "Our reaction was superb," Martínez said. "We had 25 attempts and 11 corners but it doesn't matter that their goalkeeper had an outstanding afternoon, we could have made a better pass or decision."

The contest, perhaps Sunderland's season, turned on the wrong pass and decision by Everton. Martínez made one change to keep things fresh over the festive period, Leon Osman for the in-form Ross Barkley, and it was to have damaging consequences. Everton made a sluggish start and Howard was almost sold short by a Phil Jagielka header as early as the second minute before he opted to play a goal-kick to Osman despite the close attentions of Ki. That was the first mistake. The second was a poor first touch from the Everton midfielder that gifted possession to the South Korea international, who rounded Howard but was clipped for the clear red card and penalty that duly followed.

The reactions of Osman and Howard were telling, and both their afternoons were over as the midfielder gave way to the substitute keeper, Joel Robles. Ki, scorer of Sunderland's Capital One Cup quarter-final winner against Chelsea, kept his composure to beat Robles convincingly from the spot. "We have very clear concepts about when the pass is on," said Martínez. "And when you receive the ball it's another decision. We got both decisions wrong. My only complaint is a law that means you get a double punishment, but the referee was spot on."

Only West Bromwich Albion have survived in the Premier League after being rooted to the bottom on Christmas Day and Sunderland, naturally, were receptive to the gift. Poyet said: "Football is incredible. You never know what is going to happen and everything changed with that mistake. We took advantage but, even with 10 men, we had to defend for our lives."

Martínez immediately switched to a 4-3-2 with Kevin Mirallas joining Romelu Lukaku in attack, before Mirallas disappeared down the tunnel for a toilet break for four minutes to leave Everton with nine men. "He had a stomach problem and had to go and see the doctor," explained the Everton manager, who will be without Howard and Gareth Barry for Sunday's game against Southampton after the influential midfielder received his fifth booking of the season.

The jolt did nothing for the initial quality of Everton's performance until Barkley was introduced for the second half and Sunderland, lively throughout on the break, almost doubled their lead when Ondrej Celustka tested Robles with an angled drive. Everton's keeper spilled the shot but made a superb reaction save to deny Sebastian Larsson on the rebound.

Modibo Diakité, part of a Sunderland defence missing the injured John O'Shea and the suspended Wes Brown, squandered a clear opening at the start of the second half and Steven Fletcher went close with a back-post header from Lee Cattermole's inviting cross.

But there was greater urgency from Everton's 10 men after the interval and Barkley's willingness to shoot on sight brought an added threat.

Mannone produced a fine save to keep out Phil Jagielka's header from a Bryan Oviedo corner and an even better stop when the Costa Rica defender cut inside and sent a rising shot towards the top corner from 20 yards. When Barkley attempted to reproduce the free-kick expertise that defeated Swansea City on Sunday, the Sunderland keeper thwarted Everton yet again with his finger tips.

He was finally beaten by Nikica Jelavic's diving header from the resulting corner but the on-loan Liverpool striker Fabio Borini cleared on the line, and he was spared again when Lukaku failed to meet an inviting cross in the final seconds.

"This result has to change our mentality," said Poyet, who has seen his team beat four of the top six Premier League sides in all competitions this season only to drop points repeatedly against the lesser lights. "There are no more excuses."