The Goodison club had been discussing terms with the Spartak Moscow winger for next season when the Russians said they were willing to let him go immediately as he had been virtually frozen out following a row with manager Valery Karpin.
Martinez, an admirer of McGeady when he was Wigan manager, said: "We were talking to Aiden and we agreed for him to come next season.
"But because of his situation in Spartak, we were quite happy to make it happen now. In a matter of hours it fell into place.
"I was delighted to have Aiden at the end of the season, but now it is a real boost to be able to work with him straight away."
The imminent arrival of the 27-year-old Republic of international player will now hasten the departure of former Rangers striker Nikica Jelavic's move from Everton to Hull.
McGeady was paraded to supporters at Goodison Park prior to yesterday's game against Norwich.
The transfer confirms McGeady's return to Britain four years after he quit Celtic where he was named the club's young player of the year three times.
During his time at Parkhead, the player racked up four Scottish Premier League titles, two Scottish League Cups and two Scottish Cups and won the SPFA's Players' Player of the Year prize in 2008.
His form made him one of Scotland's hottest young properties. Spartak signed him in a £9.5m transfer in 2010, and he scored 13 goals in 93 games in Russia, as well as helping them finish second in the league in 2011-12.
However, his stay in Russia has been dogged by disagreements with club officials. He has not played since November after falling out with Karpin and was made to train with the reserves.
He will now join the Everton squad for a week's warm-weather training in Tenerife ahead of their next game at West Brom on January 20.
Martinez added: "He hasn't played in the last five or six weeks. That is going to give us a little bit of time. But everything is signed and he will travel with us tomorrow to warm-weather training."