Mauricio Pochettino's side won for the first time in seven Premier League outings on Boxing Day as they overturned Cardiff City 3-0 in what was ultimately Malky Mackay's last game at the Welsh club.
Southampton also kept a first clean sheet in 10 games in all competitions at the Cardiff City Stadium to boost their confidence ahead of Sunday's trip to Goodison Park, although Everton will be hoping to atone for their festive efforts.
Roberto Martinez went into Thursday's clash with Sunderland as the only visiting manager to have won at Goodison in 2013, when he led Wigan Athletic to a 3-0 FA Cup triumph in March.
But the Spaniard could only watch in in frustration as his Everton side fell to defeat to the Premier League's bottom club courtesy of Ki Sung-Yueng's first-half penalty, which also resulted in Tim Howard's dismissal.
He said: "Joel is a young man who has been very professional in waiting to help the team and now he has an opportunity to do that.
"The whole squad have been working with that mentality since day one - being ready and being supportive of each other and clearly we couldn't be happier with the goalkeepers in the team.
"In the same way, we are missing Gareth - another experienced footballer who has set high standards. We will miss him but it becomes an opportunity for someone else. And that's where you measure the success of a team over 10 months."
As well as the suspended duo, Everton are also without Gerard Deulofeu (hamstring) and Darron Gibson (knee), but Antolin Alcaraz is back to full fitness and Martinez retains hope of including Leighton Baines after a foot injury.
The sides last met on January 21 in Pochettino's first game after replacing Nigel Adkins at St Mary's Stadium, drawing 0-0, and Southampton have not beaten the Merseysiders since September 2002.
