The two clubs agreed a deal earlier this week that will see Arthur leave Camp Nou for Turin for around €70 million (£63m/$79m), while Pjanic will move in the opposite direction for a €60m (£55m/$67m) fee.
Both players underwent medicals at their respective new clubs on Sunday afternoon before putting pen-to-paper on their contracts.
Arthur has signed a five-year deal with Maurizio Sarri’s side, which will see him earn a basic salary of €5m per season, going up to €7m (£6m/$8m) with bonuses.
Though both players will remain at their current clubs for the remainder of the extended 2019-20 campaign, Barcelona wanted the deals completed by the end of June for budgetary reasons.
Given the financial uncertainty caused by the coronavirus pandemic, Juve would struggle to pull off an expensive transfer so needed to agree a swap deal that would include one of their high earners heading in the opposite direction.
The Catalan giants have been targeting Pjanic for some time, but the deal was held up as Arthur was initially reluctant to make the move to Italy.
However, a breakthrough was reached on Saturday evening that allowed the clubs to press ahead with the double deal.
Arthur has made 72 appearances for Barcelona since joining from Gremio in the summer of 2018. The Catalan giants viewed the midfielder as a key part of their first-team squad, but Goal revealed last month they would be open to selling the 23-year-old should the right offer arrive.
Pjanic, meanwhile, has been a regular in the Juventus line-up, making 36 appearances in all competitions this season for the Turin giants, who are on the brink of a ninth successive Serie A title.
They currently sit four points clear of Lazio at the top of the table with 10 games remaining.