Jackson, 22, scored four goals in 36 appearances last season as Wrexham finished eighth in the table.
He is Barnsley's third signing for next season, ahead of their League One play-off final at Wembley on Sunday.
"Kayden has pace to burn, likes to get at defenders and will fit in nicely to the way we play," caretaker boss Paul Heckingbottom told the club website.
Jackson turned down a contract offer from Wrexham for next season.
Find all the latest football transfers on our dedicated page.

Canadian Roger Thomas Clark is said to have been a key adviser for Silk Road creator Ross Ulbricht.
The US Department of Justice alleged that Mr Clark advised Ulbricht about the best way to run the site and how to evade the police.
The Silk Road website was shut down in late 2013 following raids by the FBI and other law enforcement agencies.
It was a marketplace sited on the dark web through which many people bought illegal drugs. In May this year Ross Ulbricht was sentenced to life in prison for creating and running the site.
The DoJ alleges that Mr Clark was a "high-ranking" operator on the Silk Road and was instrumental in helping Ulbricht run it. He gave advice about ways to improve the technology underpinning the site, boost sales and on the best way for Ulbricht to hide his real identity, said US authorities.
Mr Clark was paid "at least hundreds of thousands of dollars" for this advice, said the DoJ in a statement announcing the arrest.
"Clark may have thought residing in Thailand would keep him out of reach of US authorities, but our international partnerships have proven him wrong," said FBI assistant director Diego Rodriguez.
On the site and in other underground forums, Mr Clark is believed to have used several nicknames including "Variety Jones, "VJ", "Cimon" and "Plural of Mongoose".
Extradition proceedings have been started against Mr Clark to transfer him from a jail in Thailand to the US.
He faces charges of narcotics conspiracy and money laundering. If found guilty of both charges he could face 30 years in jail.

The "very confused" pensioner was stopped by police up on the 70mph-limit road near Crawley, West Sussex, on Friday.
Officers were alerted by a motorist who had seen the man heading away from the M23 on the A2011 Crawley Avenue and had stopped to help him.
PC Katie Breeds, of Sussex Police, said: "He didn't know where he was.
"We sat him in the back of our police car and waited for a colleague in a van to collect the mobility scooter and then we delivered scooter and driver back home to his warden-assisted accommodation in Crawley."

The 24-year-old was released by the Saddlers in May after 19 appearances in three years and has now joined their League One rivals.
"My main aim is to rectify what happened at Walsall," MacGillivray told the Shrews' official website.
"I didn't play as many times as I wanted to and that's what I want to do here, playing as many times as I can."
MacGillivray, who has also played for Stalybridge Celtic and Harrogate Town, is Paul Hurst's seventh signing of the summer.
Defender Zak Jules, midfielders Jon Nolan and Ebou Adams, winger Daniel James and forwards Arthur Gnahoua and Lenell John-Lewis have also been added to the squad.
Find all the latest football transfers on our dedicated page.

It is the first professional contract for the former Bristol Rovers trainee.
"When my agent told me that Forest Green were interested I was totally thrilled," Davies told the club website. "I can't wait to get started."
Forest Green - who finished second in the National League in 2015-16 - gave a first senior deal to 18-year-old striker Olly Mehew last Thursday.
Barnsley have signed striker Kayden Jackson from National League club Wrexham on a two-year deal.
[NEXT_CONCEPT]
A man alleged to have helped run the notorious Silk Road drug marketplace has been arrested in Thailand.
[NEXT_CONCEPT]
A 92-year-old man was found riding his mobility scooter on a busy dual carriageway after taking a wrong turn.
[NEXT_CONCEPT]
Shrewsbury Town have signed former Walsall goalkeeper Craig MacGillivray on a one-year deal.
[NEXT_CONCEPT]
Forest Green Rovers have signed Wales Under-19 international striker Blake Davies on a one-year deal.