The Foxes secured a place in the third round via penalties at St James' Park on Wednesday night after a 1-1 draw.
James Maddison had put the Foxes ahead in the first half but Yoshinori Muto levelled matters for the hosts soon after the restart.
Schmeichel saved two spot kicks as Leicester came through the shoot-out 4-2, and he was delighted with the way his team-mates handled the pressure.
The keeper admits there are plenty of areas where Leicester can improve but hailed their mental strength as they set up a third round meeting with Championship outfit Luton.
"We weren't really slick in our passing, we didn't really create too many chances so there are lots of things for us to improve on, but what's amazingly good is our attitude and our bottle to go up and take penalties," he told the club website. "That's big pressure, so all credit to the lads."
Schmeichel was a member of the Leicester side that won the Premier League in the 2015-16 season and their record in cup competitions since has been solid, but they are yet to add any further silverware to their trophy cabinet.
The Foxes reached the quarter-finals in the Champions League the following season and have contested three other last eight ties since (two in the EFL Cup and one in the FA Cup).
Leicester were knocked out of the EFL Cup by eventual treble winners Manchester City last season but Schmeichel still sees that defeat as an opportunity missed - and he hopes they can challenge for the title this year.
"This is a trophy that we feel last year we should've gone further in and a trophy we ultimately should be competing for," said the Denmark international stopper.
Leicester are back in action in the Premier League on Saturday as they entertain Bournemouth and Schmeichel feels it is crucial they pick up all three points to ensure they go into the international break in the top four.
He added: "Saturday is huge for us. If we can win that game, we can stay in that top four then that's a really, really good start and something to build on for after the international break."