The 30-year-old Colombia international only joined the Premier League club last September and initially proved to be a major hit under old boss Carlo Ancelotti.
Rodriguez scored six goals in 26 appearances for the Toffees' last season, but Ancelotti's surprise exit to Real Madrid this summer saw a change in his situation at Goodison Park.
New manager Rafael Benitez told the South American he was not part of his plans and after he travelled to the Middle East at the weekend to finalise a move with the transfer window in the region still open until the end of the month, a permanent switch has now been confirmed.
Speaking after Tuesday's Carabao Cup loss at QPR, Benitez suggested the player reportedly on £200,000 a week had to leave due to financial reasons.
He said: "Don't forget, we are talking about a special situation from this club, the financial fair play rules have to manage everything in the best way we can."
Benitez had expected more information in the coming days following the cup loss in the capital and Rodriguez's departure brings an unwanted saga to an end.
When summer moves to Porto and Turkish outfit Basaksehir failed to materialise, the former Liverpool boss conceded it may be difficult to offload the club's highest earner.
But Al-Rayyan will be the next destination for Rodriguez, who has failed to hit the heights achieved at the 2014 World Cup where he won the Golden Boot.
He has represented Porto, Monaco, Real Madrid, Bayern Munich and Everton in Europe, producing a number of eye-catching displays at the beginning of his tenure in England before his form and the club's tailed off in the second half of the previous campaign.
Benitez has enjoyed a strong start to life at Goodison Park but one of their shots at silverware are over after a penalty shoot-out loss at QPR in the Carabao Cup.
Following the 2-2 draw, Tom Davies had a spot-kick saved by Seny Dieng to give the hosts an 8-7 victory on penalties at Kiyan Prince Foundation Stadium.
Spaniard Benitez called for improvements in defence, adding: "The most difficult thing to improve, normally, is in attack because it depends on the quality of players, but we are creating chances, so I am sure we will score goals.
"But in defence we have to be more aware of what is going on around and be sure we defend as a team, and then (if) we are strong enough in defence, that will give you the possibility to win games.
"You have to build the momentum from a solid team. At the same time, we are creating chances, so we have to find this balance that still we cannot find."
QPR boss Mark Warburton spoke up for the Carabao Cup following the tie.
He told his club's official website: "I spoke to a couple of people and they said 'you want to be in that thing?' and I cannot for the life of me understand it.
"Anyone would know the importance of going as far as you possibly can in every major competition. It's a magnificent tournament.
"We've got a Premier League scalp, a packed stadium, the fans were magnificent, getting behind the team, more belief in the team, they see what it means to the players that pull the jersey on. All the old cliches come out and there's a reason why they are old, they are around for a reason.
"You saw everything about a magnificent cup tie, a great stadium, the noise, the smell of the grass, the floodlights - outstanding, and that's why we play football."