Steven McDonnell put the Tigers ahead on the half-hour mark with a superb header before Lynton Karkach added a second on 54 minutes as McDonnell switched play to Karkach, whose first sublime took it past his marker to fire under the keeper.
Benjamin Rhodes then wrapped things up with a third in the final minute of the game with a low drive.
“They got the job done in the end,” said manager Kyle Jordan.
“There were times in the game when we were a little bit slow. We looked a little bit lethargic and moved the ball a bit too slow.
“But in all probability that was probably credit to Ecclesfield.
“I thought they worked really hard. They put themselves about and made it difficult for us to break them down.
“In truth I thought we were really poor first half – not effort-wise but just a lack of quality on the ball. You could probably tell we’d not been able to play for a while because of the weather.
“But they go the job done so I can’t grumble too much. I was pleased overall.”
Ecclesfield managed to park the bus for the first half hour to frustrate the home side until they finally made the breakthrough.
“We thought that might happen – similar to how we went to Whitby,” said Jordan.
“We went with a game plan to try to stay in the game as long as possible and keep them out.
“It was very effective how they did it last night – especially in the first half.
“Harry (Holland) has been brilliant for us every time he’s played and always looked really comfortable on the ball.
“But last night his passing wasn’t there until the second half, when he really picked it up.”
Boss Jordan was happy to hand Deegan Atherton the captain’s armband for the Ecclesfield game and it paid off with a man of the match display.
“We had a few young lads in the team along with the more experienced ones tonight and we though he was a good blend and a good link between the two,” said Jordan after the game.
“Every time he plays for us he is brilliant. He very rarely has a bad game.
“He is honest, he listens to what you say and he takes it on board. I thought he was our best player tonight – head and shoulders above everybody else.”
He added: “I said to the boys in the dressing room before the game that we wanted them to play how they would on a Saturday in a league game. We don’t want people doing things they wouldn’t normally do on a Saturday.
“He is one you can bank on to do that. He did everything well, he defended well, won his headers and won his tackles.
“He stepped up to it big time and I am pleased for him.”
Wednesday’s win took the Tigers into the quarter-finals and Jordan said: “Everyone likes to get into a cup final and it’s brilliant for clubs with a big fan base like us.
“It’s something to potentially look forward to. But it’s the quarter-finals next so there are two tough games to look forward to first.”