John Sheran admitted Cove dug themselves a hole but was pleased with their response against Lossiemouth.
His side won 4-2 thanks to two late goals from Paul McManus and Mitch Megginson.
But they were made to work hard by a feisty Coasters side at Harlaw Park.
Cove were twice pegged back by new manager Bobby Beckwith’s side before securing victory late on.
And despite the difficult manner of victory, Sheran praised his side for the resilience shown to get to the quarter-finals.
He said: “We didn’t panic when it went to 2-2.
“The guys were experienced enough to say ‘that’s where we are, now let’s get ourselves out of this hole’ and they did.”
He added: “Lossiemouth take a lot of credit, they played really well, caused us lot of problems and got back to 2-2 but we never really looked like losing the game.
“It’s good to be through and into the next round of the cup and that’s all that matters for us now.”
McManus scored after eight minutes but the visitors equalised through Ross Archibald 14 minutes later.
Megginson restored Cove’s lead with his first of the game just before the half-hour mark.
Just less than 10 minutes after half time, Chris Ross levelled the scores once more.
But with just five minutes left, McManus notched the vital third goal before Megginson sealed the game.
“They’re both good strikers,” Sheran said of his new forward tandem, “The team in general creates a lot of opportunities for the players up front.
“It was good they both got goals and it builds a bit of confidence.”
The result sets up a quarter-final trip to Christie Park to face Huntly.
But Sheran refuses to consider the possibility of winning the cup at this early stage in the competition.
“In cup games, you just go onto the next one and try and win that one,” he said.
“If you start thinking too far ahead, there are too many good teams that can beat you.
“From our point of view, it’s Forres in the next round and that will be a tough game so we can only focus on that.”