Källström, who joined the London club in a January loan move, came on in extra-time after Arsene Wenger's team had cancelled out Jordi Gomez's second half penalty.
"It was a really tough game, they defended really well but we pushed all the way through," the Swede said.
The match looked like adding to the slew of disappointments to befall Arsenal in recent weeks until Germany centre-back Mertesacker stooped to head in Alex Oxlade-Chamberlain's scuffed shot with just eight minutes remaining of regular time.
The Gunners held their nerve to win the penalty shoot-out 4-2 following the 1-1 draw after extra-time.
"The semi-final is all about winning. It was tough but we did it. We had to stay strong mentally and at 1-0 down it gets tough. We were focused and even though it was on penalties, we will take it," Källström said.
Lukasz Fabianski was the hero as he saved Wigan's first two penalties, turning away efforts from Gary Caldwell and Jack Collison, while Källström, Mikel Arteta, Olivier Giroud and Santi Cazorla all converted to send the Gunners through.
The north Londoners, whose last trophy came in the 2005 FA Cup, will now face Hull City or Sheffield United in the final on and can spend the next few weeks focused on the battle to qualify for the Champions League.
This was an unconvincing display from Arsenal for long periods but German giant Mertesacker hopes the gritty way his team ground out victory provides a much-needed morale boost.
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"In the first half we were a bit timid. In the second half I think we hit the post twice and had a couple of chances so it was a good comeback," Mertesacker, who had conceded the penalty which Gomez converted, told ITV.
"I think we have great character, even though a lot of people have questioned that."
"We have been struggling in recent weeks but you have to deal with that, it's part of the game."