Bassala Sambou’s 48th-minute strike proved decisive as the young Blues overcame a strong Foxes outfit, who finished third in Premier League 2 last season.
And while Unsworth admitted his side’s first half performance did not hit the standards he demands, he was quick to praise his players’ quality and application after the break.
“It was a great win," he said. “A great second half performance and a great reaction against a very good team.
“In the second half, we needed to start playing, believe in ourselves and play on the front foot, which we didn’t do in the first half.
“In the second half, we were very much like ourselves. We were terrific, we started brightly, we got a great goal. We passed the ball, we won headers and tackles. We were like us.
“As soon as we got the goal, the belief was there and the confidence grew from that. I’m very proud of the second half display.”
The Foxes started the quicker at Goodison and Kieran Dewsbury-Hall almost gave them a perfect start, his fizzing 20-yard effort going just wide on four minutes.
Leicester continued to look the more threatening in the first half-hour, without carving out any clear-cut chances.
They did put a few teasing balls into the box but each time Toffees goalkeeper Joao Virginia – on his Everton debut - confidently plucked the ball out of the air.
Indeed, the Portuguese’s composure and bravery were evident throughout an eye-catching Blues bow.
Everton gradually gained a foothold in the game and almost found an opening when Brendan Galloway burst down the left-hand side, but his cutback was intercepted with Fraser Hornby waiting to pounce.
The young Blues' best chance of the first half was to fall to Sambou. On 35 minutes, Nathan Broadhead found some space on the edge of the box and played a perfectly-weighted pass to the striker but his shot went fractionally over.
Everton flew out of the blocks after the half-time break and on 48 minutes Sambou did get on the scoresheet - and in some style.
Sambou showcased impressive strength to hold off a Leicester defender, then found Joe Williams who drove at the Foxes backline before playing an inch-perfect return pass through to the former Coventry City man, who expertly lifted the ball over ‘keeper Viktor Johansson.
The goal buoyed the young Blues and three minutes later Antony Evans almost doubled their lead, his powerful, swerving free-kick looked destined for the top corner before the Leicester ‘keeper produced a sprawling fingertip stop.
Evans was denied by Johansson again minutes later when his low drive was tipped round the post.
Leicester gradually regained their composure but Everton continued to look dangerous on the break. Fraser Hornby shot agonisingly wide on the hour mark after Galloway played him through on goal.
The closest the Foxes came to snatching an equaliser came four minutes from time, when substitute Ryan Loft’s deflected shot from 20 yards brought out a brilliant reaction stop from Virginia.
Boss Unsworth reserved special praise for the 18-year-old Portuguese goalkeeper’s performance.
“He was absolutely terrific,” Unsworth said. “He’s set a standard and a benchmark at this level with his performance tonight.
“For a such a young goalkeeper, I’ve haven’t seen a performance like that at this level.
"It wasn’t just his saves, it was how commanding he was from set-pieces, from crosses, vocally he was loud and giving good information. He kicked it really well, to, and he was composed. He looks like a great signing.”