The hosts started brightly enough and Radoslaw Majewski's mistimed challenge on Stephen Dawson afforded them a free-kick on the edge of the box, but Marlon Harewood fired over the crossbar against his former club.
Forest responded immediately as woeful defending from John Stones gifted the ball to Andy Reid but the midfielder, making the 300th league start of his career, lifted a tame shot into the hands of Ben Alnwick.
The visitors were made to pay midway through the half when Keith Hill's side constructed a superb counter-attacking goal.
Dawson burst forward from the edge of his own box, laid the ball off for Jacob Mellis and his precise pass set Harewood away to round Lee Camp and roll home.
Despite the earlier taunting Harewood, who returned to Forest for a goalless six-game spell last season, refused to celebrate in front of the travelling fans.
He almost set up a second within 60 seconds, only for Brendan Moloney to clear away his dangerous cross with Matt Done lurking.
And yet by half-time, Forest were 3-1 up. First Halford cut inside and though his shot with the outside of his right boot was weak, a deflection off Scott Wiseman left Alnwick wrong-footed as the ball rolled into the net.
Then four minutes before the interval, Cohen beat Scott Golbourne in a one-sided footrace for a loose ball, squared for Billy Sharp and when Alnwick could only parry his effort, Cox was on hand to tap in the rebound.
Cohen's goal, right on the stroke of half-time, was the crowning glory of a phenomenal 10 minutes for the visitors. He set his sights to shoot from 25 yards and sent an unstoppable effort into the top corner.
Hill brought on Kelvin Etuhu, Chris Dagnall and Tom Kennedy early in the second half as he sought to reinvigorate his flagging charges.
Kennedy forced a smart near-post save from Camp with a low, driven free-kick from the edge of the box on the hour mark and Dan Harding cleared well from Dagnall's cross, but it was a struggle for Barnsley to get a foothold.
When Etuhu connected with Dagnall's flick-on in the six-yard box, only to be denied by a sprawling Camp save, their best chance of the half was gone.
Jenas, introduced eight minutes earlier, raced through to add the fourth after 78 minutes, producing an accomplished chip over the helpless Alnwick.