But their hopes were ended as the Black Cats, the division's bottom side, emerged with a surprise 1-0 away win.
Jagielka admits there is disappointment in the dressing room, but has stressed what is really important is a team's achievement over the course of a season.
And the 31-year-old England defender has also suggested that with the 2013 home league record target now no longer on the agenda, fifth-placed Everton may actually head into the encounter against ninth-placed Southampton feeling as if some pressure has been lifted off them.
He told evertontv: "We're very disappointed but the season is August to May, not a calendar year, and we'll get over that now.
"Maybe that's a little monkey off our backs and we can go out and play well on Sunday, get those three points and climb up the table."
A 25th-minute penalty from on-loan Swansea midfielder Ki Sung-yueng helped steer Sunderland at Goodison Park to what was their first victory in seven league matches, and maiden away win of the season.
The spot-kick was awarded after Leon Osman miscontrolled a short goal-kick from Everton goalkeeper Tim Howard.
Ki latched onto the ball and darted into the box, where he hit the turf when tackled by Howard, who was shown a straight red card.
The Toffees - unbeaten in 10 games prior to the fixture - piled on the pressure after the break but could not find an equaliser, with Black Cats stopper Vito Mannone pulling off several impressive saves, including a diving effort to keep out a free-kick from Ross Barkley.
Assessing the game in isolation, Jagielka said: "Obviously we didn't play very well in the first half and sort of shot ourselves in the foot with the goal we conceded.
"But it shows the character in the dressing room that we created quite a few decent chances and unfortunately for us we came up against a goalkeeper in some fantastic form. We certainly gave it a good go.
"Even after Mannone saved the free-kick from Ross Barkley, I still felt we could create a couple more chances or half chances, which we did.
"But we just didn't manage to get the ball over the line and in that the boys are disappointed."
As well as the suspended Howard - set to be replaced on Sunday by Joel Robles, who came on for his Everton league debut against Sunderland - the Toffees will also face Southampton without Gareth Barry, with the on-loan Manchester City midfielder having to serve a ban after picking up his fifth booking of the season in the Boxing Day game.
Jagielka is confident Everton will be able to cope with those losses.
"It's pretty simple," he said.
"Joel will obviously come in for T-How. He's been waiting for his chance and he's played well in some of the cup games.
"The manager (Roberto Martinez) will then decide what he'll do with the midfield, how he'll rejig that.
"But we've got a decent squad and some good players to come in, some fresh players as well - which is obviously important over this period - so hopefully they can come in, freshen up the squad, and we can put in a good performance on Sunday."
Forward Kevin Mirallas was substituted at half-time during the Sunderland contest having felt unwell and it remains to be seen whether or not he or Toffees full-back Leighton Baines, who has been sidelined by a broken toe but nearing readiness for a comeback, will feature.