The former Spain midfielder, who announced his international retirement earlier this week, has also now called time on his five years at the Bernabeu to join German giants Bayern in a transfer reportedly worth 10 million euros ($13.28million).
Alonso underwent his medical with Bayern on Thursday, with the deal being completed on Friday morning.
The 32-year-old then held a farewell press conference at the Bernabeu where he admitted it was not an easy decision to leave the European champions.
The former Liverpool player, who has seen competition for a midfield berth at Madrid this summer increase with the signings of Toni Kroos - from Bayern - and James Rodriguez, said: "It's a sad day on one hand because I'm ending a marvellous spell which I've enjoyed a lot.
"It's been five great years. Now a new cycle is starting and it's been the most difficult decision of my life. I've thought long and hard about it.
"I wanted to be honest with myself and Real Madrid. You need to know when to say goodbye and this was the best time.
"I will always be part of the Madrid family."
As well as giving his thanks to the Spanish giants, Alonso explained why his next step is going to be with Pep Guardiola's Bayern.
Alonso also insisted the arrival of fellow holding midfielder Kroos at the Bernabeu had no impact on his decision making
The 2010 World Cup winner said: "It's not because he (Kroos) arrived. Madrid have made a great signing. The two of us would have played a lot. This is not a tantrum. This decision has been well thought out and evaluated.
"It was quite a recent interest (from Bayern), in the last two weeks. After winning 'la decima' (Real's 10th European Cup) I felt that I needed to make a fresh start.
"The club didn't want me to go and it was my choice, they didn't agree but they understood.
"With 'la decima' we removed a huge weight off our shoulders and after that great achievement I needed other things.
"Going to Bayern was the best option. Bayern are going to fight for all the titles. I can't wait to start."
Alonso becomes the fifth Spanish player on the German giants' books together with Jose Reina, Juan Bernat, Thiago Alcantara and Javi Martinez - who is out until the new year with a serious knee injury.
Bayern on Wednesday completed the signing of Mehdi Benatia from Roma for a fee of 26 million euros and they were not expected to venture into the transfer market again this summer.
Alonso's arrival could see Xherdan Shaqiri seek a move away from the club.
The Switzerland international is attracting interest in Italy with Juventus and Inter Milan both keen on the former Basle midfielder.