Roberto Martinez's side headed into the first Merseyside derby of the season having claimed just five points from as many Premier League matches.
A first win at Anfield in 15 years proved beyond Everton on Saturday, but Jagielka at least ensured his side did not leave empty-handed, cancelling out Steven Gerrard's 65th-minute free-kick with a powerful long-range strike in stoppage time to end it 1-1.
Having rescued an unlikely point for the visitors, Jagielka urged Everton to build on the result in their upcoming fixtures.
"Hopefully next time we come to Anfield we can go one better but a point is not the end of the world," the England defender told BT Sport.
"It has been a frustrating season so far; to go 1-0 down and keep knocking at the door showed tremendous character. We need to keep showing character.
"The lads need to keep playing the same way, keep having the courage to do the right things and hopefully this will be a kick-start to the season for us."
Everton manager Martinez also hailed the spirit of his players.
"One point in the manner it arrived [can] change the dynamic of our season," added the Spaniard.
"I'm so satisfied with the character that we showed.
"We didn't accept defeat. I think it's a real example of the character and the DNA that we have in the club."
Everton face Manchester United in their next Premier League fixture a week on Sunday, which follows a trip to Krasnodar in the UEFA Europa League.
‘After Manchester City’s recent form, maybe they’re the underdogs against Manchester United!’ Former Red Devils defender on this weekend’s derby
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
