The actor and TV presenter beat finalists Louise Minchin and Jimmy Osmond after a final challenge and tense cook-off.
The BBC One show was watched by 4.9 million viewers, peaking at 5.7 million, giving the show a 26.1% audience share.
After his win, Conran said: “I can’t believe it, I have been working up to this for so long, for this one moment. I am so very happy and relieved. My family are going to be so proud.”
He added: “I was very happy with ‘finalist’ but I’ll take ‘champion’.
“’Champion’ sounds great.”
After four weeks of heats and two weeks of gruelling semi-finals and finals, judges John Torode and Gregg Wallace whittled 19 celebrities down to three.
Newsreader Minchin, singer Osmond and TV presenter Conran had to take on the Chef’s Table challenge, hosted by Michelin starred chef Atul Kochhar.
They were asked to impress some of the UK’s top chefs with their creations, including Cyrus Todiwala, Alyn Williams and Ed Baines.
Following this, their last task was to cook a three-course meal for Torode and Wallace.
Conran impressed with his octopus stifado and fava starter inspired by his upbringing in Greece, followed by a main dish of stuffed chicken leg with feta curd, baby globe artichokes, peas and beetroot.
His nut filo pastry pinwheels with yoghurt mixed with condensed milk and fig also drew praise.
Wallace said: “Alexis is a very, very worthy Celebrity MasterChef winner. Aligning that wonderful presentation with his Greek heritage has resulted in some of the best food I have ever seen in this competition.”
Torode said: “Alexis’s dishes looked amazing. His food has become sophisticated, exciting - it is just fantastic to eat. I loved it.”
The show’s 2016 champion will join a line-up of previous winners including Kimberly Wyatt, Phil Vickery, Lisa Faulkner, Nadia Sawalha and Matt Dawson.
He said: “The key for the final for me was cooking from the heart, as that is what cooking is all about.”