The teams were level at the break, but, after former Grecian Jenna Markham opened the scoring, Barron-Clark scored first from a tight angle before finding the back of the net from 30 yards.
That saw the Cherries get themselves in the hat for the last eight of the competition.
Three changes saw Helen Bleazard recall Markham to face her former club, while Katie James and Katie Scadding also started in place of Georgia Wilson, Amber Treweek and Erin Foley.
Bournemouth started well in Newton Abbot.
Jenna Markham scored against her former club _(Image: Richard Crease)_
Kenni Thompson found Barron-Clark after seven minutes and the attacking midfielder tested keeper Olivia Giles, before Jess Hennessy turned a Thompson corner just wide midway through the half.
Barron-Clark went close with a shot that just clipped the post, before Exeter threatened, Scadding denying Sophie Gillies.
The teams went into the break with neither able to break the deadlock, but it didn’t take long after the resumption for Bournemouth to get their noses in front.
Sophie Quirk was the architect as she created space to cross for Markham, who, unmarked, was able to guide her header into the bottom corner after 55 minutes.
The hosts were then reduced to ten players as Jaydee Seaman was sent off for a second bookable offence just after the hour mark.
From there, the Cherries pressed home their advantage.
Molly Barron-Clark _(Image: Richard Crease)_
Barron-Clark raced onto a through ball from James and turned to curl the ball in at the far post for her first goal of the afternoon.
She struck the cross bar just minutes after her first goal, before confirming the win for Bournemouth with a brilliant goal from 30 yards fired into the back of the net to complete her brace and a 3-0 win for the Cherries.
The clean sheet means Bournemouth are yet to concede a goal across five games in the competition, and they are in the hat for the draw tonight, with the quarterfinals set to take place on the weekend of January 11.