In a statement, Juve said they had agreed with Atalanta a fee of "51.3 million euros, payable in four financial years, with the addition of ancillary costs up to 3.4 million euros".
Juve will pay Atalanta, "upon the achievement of further performance objectives", another six million euros for Koopmeiners, who has signed a five-year contract worth a reported 4.5 million euros a season.
It is the biggest deal of the summer in Italy, beating the 51.5 million euros Juve agreed to pay to bring Douglas Luiz to Turin from Aston Villa.
Koopmeiners' arrival signals an end to the highest-profile transfer saga of the Italian summer transfer window, with the Dutchman pushing for a move away from Atalanta.
Atalanta dropped Koopmeiners from their squad for the first two games of the season as one of the stars of May's Europa League triumph reportedly handed in doctor's notes for stress while the club initially held out on selling to Juve.
But the two clubs eventually came to an agreement and Juve have another signing for new coach Thiago Motta as the "Old Lady" of Italian football undergoes a makeover.
Koopmeiners was one of the best players in Serie A last season, scoring 15 times in all competitions as Atalanta won their first major trophy since 1963.
Juventus are currently top of Serie A after winning their first two matches against Como and Verona, and host struggling Roma on Sunday.
Atalanta announced almost immediately afterwards the signing of Ivory Coast international Odilon Kossounou on loan from Bayer Leverkusen with an option to buy.
Italian media report that Atalanta have agreed to pay German champions Leverkusen 5.5 milion euros for the season-long loan and can make the move permanent for 25 million euros.