The Mariners passed up several first-half opportunities but Conor McKenzie drilled in the opener on 53 minutes and that seemed to settle any nerves.
A crowd of around 300 witnessed further goals from substitutes Tiwi Daramola and Connor McMullan, as Camelon ran out comfortable winners in the end.
Mariners boss Gordon Herd told the Falkirk Herald: “It was a bit of a nervy first half. Had we got the early goal, I think it would have been different. It was big day for the club and I think the occasion might have got to a few of the boys, but that’s something they will learn from.
“We were also the favourites, so that brings its own pressure. Newton Stewart tried to frustrate us but, if we had taken our chances, it could have been five or six in that first half.
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“I thought the substitutes came on and made an impact. Tiwi and Connor did well. It’s a great strike from Conor for the first goal, he deserves that. He goes about his business quietly.”
They will face either Auchinleck Talbot or Dundonald Bluebell, who will battle it out in a replay after a 2-2 draw.
Herd said: “The carrot is there for the players to force themselves into that starting XI in a couple of weeks’ time. I’m not fussed who we get but it will be a hard game regardless.”
It’s back to league action for Camelon this weekend when they travel to Prestonfield to take on the early pacesetters Linlithgow Rose, who’ve won their opening three games. The Mariners are still looking for their first league victory.
Herd said: “We were supposed to have a free week but this will stop us having fixture congestion and keep us ticking over. It’s a tough tie. Linlithgow have started really well but we’re going to go there and see what we can do. We’ll make a game of it; we won’t shut up shop.
“We firmly believe we’ve got boys in there who can get a result. It’s hasn’t been the easiest of starts, with Broxburn, Jeanfield and Linlithgow in the first four games. It could be a game to kick-start our season but it’s not the be-all and end-all. We’re not panicking; it’s still only August.
“We feel we are in a false position and that will even itself over the season.”
Jason Deans, Ross Cameron and Alan Sneddon all come back into contention after missing last week.