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The Portuguese champions revealed that the deal was worth a combined fee of €70 million.
Monaco, who will be back in the French top flight next season after winning the second-tier title, said 26-year-old Portuguese international midfielder Moutinho and 21-year-old Colombian international forward Rodriguez had both agreed five-year deals.
Porto said Rodriguez was valued at €45 million and Moutinho at €25 million, with the deal "the biggest in Portuguese football history."
Porto will make a massive profit on both, with Moutinho costing €11 million when he signed from Sporting in July 2010.
Rodriguez signed the same month from Argentine club Banfield for €5.1 million.
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.
Monaco, backed by Russian billionaire owner Dimitri Rybolovlev, currently benefit from the favourable tax laws in place in the Mediterranean principality.
Foreigners playing for the club do not pay income tax, while French players have paid less in national insurance contributions than they would at other clubs in France.
A decision taken in March by the French football league (LFP) means Monaco will be subject to the same tax laws as other French clubs from June next year, although the club themselves have promised to fight the new rules.
Claudio Ranieri's side have also been heavily linked with a €60 million move for Atletico Madrid's Colombian striker Radamel Falcao.
