Goals either side of the break from Danny Graham and Moore put Swansea, who rejected Manchester City's £6.2million bid for Scott Sinclair earlier in the day, firmly in control.
But the Welsh side were wobbling after Bobby Hassell pulled a goal back for the Championship visitors, until Moore slotted his second with three minutes remaining to maintain Swansea's 100% start to the campaign.
Michael Laudrup handed debuts to record signing Ki Sung yeung and Kyle Bartley as he made seven changes from the side that beat West Ham 3-0, with Swansea looking to avoid a repeat of their embarrassing exit against Shrewsbury at this stage of last season's competition.
Barnsley boss Keith Hill also rang the changes from their 5-1 thumping at Brighton but it was the Tykes who made the more solid start, and they saw plenty of the ball in the early going as Swansea's new combinations struggled to settle.
The lack of understanding for the hosts was evident when Moore's poor pass fell to Jacob Mellis, who drilled a fierce effort that Gerhard Tremmel did well to hold.
But Swansea began to find some of the rhythm that has helped them make such a strong start to the season.
Graham's header from Nathan Dyer's cross was well blocked by Barnsley skipper Scott Wiseman, but the striker had an easy finish for the opener after 24 minutes.
Angel Rangel delivered an inviting cross from the right and Moore astutely headed back across goal for Graham to chest in his second of the season from less than a yard.
Ben Alnwick, back in the Barnsley goal after suspension, tipped away a stinging drive from Dyer after excellent build-up play from Swansea, while Tomasz Cywka troubled Tremmel with a dipping volley that the German just held, clouting his head against the post in the process.
Moore, who was guilty of being wasteful in possession on occasion, should have doubled the lead 11 minutes before the break but he headed wide after Rangel again overlapped on the right.
But the former Aston Villa man made no mistake when he found the net just short of the hour.
Jonathan De Guzman found substitute Jazz Richards, and the full-back slipped into the area to find Moore, who turned and fired left-footed high into the net.
That should have been the game won for the hosts but they let the Championship side back in it as poor marking from Scott Golbourne's corner allowed Hassell to get in a header which beat Tremmel at his right post.
Tremmel then had to be alert to prevent an equaliser. Chris Dagnall looked set to get a reward for his industrious efforts when he stole the ball from Bartley and raced in on goal, but the keeper was out quickly to deny him.
The impressive Hassell then rattled Tremmel's crossbar from distance with Swansea rocking.But Moore settled any nerves by sliding home from an acute angle after Mark Gower had charged down a Stephen Dawson clearance.
|Team Name||Swansea City||Barnsley|
|Shots On Target||4||4|
|Shots Off Target||4||5|