Monaco, who will be back in Ligue 1 next season after winning the second-tier title, said Portugal midfielder and the 21-year-old Colombia playmaker had both agreed five-year deals.
Porto said Rodriguez was valued at €45m and Moutinho at €25m, with the deal "the biggest in Portuguese football history."
Porto will make a massive profit on both, with Moutinho costing €11m when he signed from Sporting in July 2010. Rodriguez signed the same month from Argentine club Banfield for €5.1m.
The deal is a sign of Monaco's intention to challenge towards the top end of Ligue 1 next season, with no other club currently able to compete with champions Paris Saint-Germain's financial might.
Claudio Ranieri's side have also been heavily linked with a €60m move for Atletico Madrid striker Falcao.