James Garner, Hackney and Thierno Barry were all on target at Hill Dickinson Stadium as Everton secured their quarter-final spot.
Hackney expressed delight at opening his goalscoring account for the club, describing the victory as an important moment in Everton's cup campaign this season.
“I’m delighted with that, it was a special night,” he told club website.
“I think us as a team and the fans knew it was a big game. After losing the last two games – I know it was against good opposition – but I feel like that was a big moment tonight.
“I think it was an important night against Premier League opposition.
“The fans really helped. They were unbelievable.”
He added: “I’m always working on arriving in the box at the right time. It was a great ball from Barry, and I managed to put it in the corner.
“It was a great feeling to get my first goal for this amazing club and hopefully there’s many more to come.”