It was Cilic's second win in six meetings with Rublev, who reached the quarter-finals of last year's Australian Open.
Medvedev capped his amazing year with a 7-6 (7), 6-2 win over Marin Cilic in the second singles match to give Russia an insurmountable 2-0 lead over Croatia and its first Davis Cup title since 2006.
The Italians were one of the favourites for the title but paid for Lorenzo Sonego's shock defeat by world number 279 Borna Gojo in the first singles rubber in Turin.
Cilic, the 2014 US Open champion, had been expected to mount a strong challenge for the Wimbledon title after winning the grass-court warm-up at Queen's Club two weeks ago.
Marin Cilic eased into the Wimbledon second round on Monday with a 6-1, 6-4, 6-4 win over Yoshihito Nishioka.
Queen's Club champion Marin Cilic moved up a spot to fifth after his win over Novak Djokovic, who moves back into the top 20.
Novak Djokovic hasn't earned a major title since completing his career Grand Slam by winning the 2016 French Open.
Marin Cilic powered through the deciding set to leave Novak Djokovic shell-shocked as he won the Wimbledon warm-up event.
Cilic, last year's Wimbledon finalist, downed the 70th-ranked Sousa 6-1, 7-5, 6-2 and will next play either American Ryan Harrison or Pablo Cuevas of Uruguay.
Ranked a humble 89, Simon landed without his coach and other support staff but was ready for the two Grand Slam finalists in 2017 at the Tata Open.
Simon, who was once a top-10 player but now placed a humble 89th, shrugged off opening set debacle to stun Cilic, the 2017 Wimbledon finalist, in one hour and 51 minutes.
Top seed Cilic from Croatia outplayed eighth seed Frenchman Pierre-Hugues Herbert 6-3 6-2 in 64 minutes while Anderson survived a mighty scare from a dogged Mikhail Kukushkin.
Jack Sock became the first American to reach the semi-finals of the season-ending event since Andy Roddick in 2007.
Roger Federer made a strong start to his bid to cap his golden year as the Swiss star defeated Jack Sock on Sunday.
World number six Marin Cilic and 20th-ranked Sam Querrey have joined the teams for September's inaugural Laver Cup.