Saul sealed his move to Chelsea past midnight after issues with the Spanish registration system, joining the Blues on a one-year loan deal.
The European champions have paid Atletico Madrid 3.4 million pounds in an initial fee, with the option to make the deal permanent next European summer for 30m pounds.
Chelsea director Marina Granovskaia added: “We welcome Saul to the club and believe he completes our squad as we compete for honours in five different competitions this season.”
Despite Saul’s exit, Atletico bolstered their ranks in a shock move for Griezmann, which laid bare Barcelona’s financial challenges.
Griezmann re-joins Atletico on a one-year loan with the option to extend for a further year.
Kylian Mbappe’s proposed move from Paris Saint-Germain to Real Madrid did not materialize, although the Parisians landed Portuguese left-back Nuno Mendes on loan.
Real landed French teenage talent Eduardo Camavinga for an undisclosed fee from Stade Rennes.
Arsenal were busy, signing Japanese defender Takehiro Tomiyasu from Bologna, while Hector Bellerin was loaned to Real Betis.
Leeds United sealed a deal for Manchester United’s Dan James, after the Red Devils completed their move for Cristiano Ronaldo.
Crystal Palace signed Celtic striker Odsonne Edouard, while Helder Costa moved from Leeds United on loan to Valencia and West Ham added Croatian Nikola Vlasic from CSKA Moscow.
Everton snapped up Salomon Rondon from Chinese club Dalian Professional and Tottenham signed Emerrson Royal from Barcelona, while Serge Aurier released by Spurs.
Elsewhere, Yacine Adli has joined AC Milan from Bordeaux for an undisclosed fee, Ilaix Moriba moved from Barcelona to RB Leipzig and Moise Kean went to Juventus from Everton on loan.