Eagles, who is starting to find the net on a regular basis, was joined on the score sheet by Chris Iwelumo as the Clarets turned on the style in a one-way second period.
He curled home his first in superb fashion from the edge of the box before dispatching his second from the spot. Iwelumo's eighth goal of the season and eighth at Turf Moor ensured Burnley's unbeaten home record remained intact.
The home side made a promising start and Eagles looked in the mood, delivering a fantastic ball into the box for Jay Rodriguez with just two minutes gone.
However, the Clarets youngster was inches away from giving Burnley the perfect start having cleverly pulled away at the back post.
The hosts were already playing some good football during the early stages of this game, but the visitors could have forged ahead in the seventh minute.
Hugo Colace dinked a lovely ball over Andre Bikey for Goran Lovre, but Leon Cort came racing back and produced a superbly-timed sliding tackle to deny the Tykes midfielder just as he looked to pull the trigger.
After Barnsley had enjoyed a good spell of possession, there was a coming together between Rodriguez, who had overrun the ball, and Stephen Foster in the box, but referee Andy Haines correctly waved away the home side's appeals.
Tyrone Mears then skipped between Jason Shackell and Jay McEveley, but having done all the hard work, the Clarets full-back failed to find one of several team-mates who has burst forward in support.
Dean Marney thumped goalwards on two separate occasions but Luke Steele comfortably dealt with both, while at the other end Lovre forced a smart stop out of Burnley goalkeeper Brian Jensen.
The home side went close again in the 30th minute when Steele dropped a deep Eagles cross, but Iwelumo's lobbed effort on the follow-up was cleared to safety by Foster.
With just three second-half minutes gone, Kieran Trippier wasted a glorious chance after Brian Easton, making his league debut, was robbed of the ball.
The Manchester City loanee advanced towards goal but dragged wide of the far post from a tight angle with Jensen closing in.
The hosts broke the deadlock in the 50th minute and it was Eagles, who oozed confidence throughout, angling past Steele from 25 yards.
Iwelumo nodded down an Easton cross, Wade Elliott had the composure to lay-off and Eagles did the rest, curling home his fifth goal of the season in fine style.
Colace would have undoubtedly got an effort on goal two minutes after but for Mears' defensive header at the back post.
Iwelumo's header from an Eagles cross was clawed onto the crossbar by Steele, before Eagles' cross struck the raised arms of Nathan Doyle in the 66th minute and the lineman flagged for a penalty.
After a slight delay while Barnsley protested the decision, Eagles beat Steele with pace and power, picking out the bottom right-hand corner.
Rodriguez then hit the post via Steele's back before Iwelumo wrapped things up nine minutes from time, drilling home after the ball broke kindly in the box.