The Mexican center back's transfer headlines a flurry of transactions, with Lucas Lobos going to Toluca and Sinha joining Queretaro.
The 32-year-old moves on a three-year deal, according to widespread reports, with La Maquina becoming Rodriguez's third club in Mexico. He has been champion with the other two: Chivas and America.
The other major move was Lucas Lobos moving from Tigres to Toluca, while Sinha went from Toluca to Queretaro.
MORE: U.S. Soccer MVP rankings | USA vs. Turkey photos
The big-name transfer involving Oribe Peralta from Santos Laguna to America went through, while Chivas added Sergio Napoles and David Toledo to other new signings Fernando Arce, Angel Reyna and Carlos Salcido.
Cuauhtemoc Blanco's move to Puebla was also confirmed.
Newly promoted Leones Negros signed 10 new players, including former Atlas captain Leandro Cufre, Ecuador international Fidel Martinez and former Chivas defender Hector Reynoso.
Almost $66 million was spent Wednesday, encompassing 109 deals, of which 38 were outright buys and 71 were loans.