The BBC news and sports broadcaster had received no dance training prior to taking part on the show and finished top of the leaderboard on Saturday night.
The pair collected the glitterball trophy ahead of favourite Danny Mac and partner Oti Mabuse and former pop star Louise Redknapp, who was partnered with Joanne’s brother, Kevin Clifton.
Oduba, who in the lead-up to the final was quoted as saying he did not even dance at his own wedding, was left speechless as he was handed the trophy.
He thanked his dance partner, production crew and judges before telling Clifton: “I love you with all of my heart. I’m so speechless. I just want to say thank you.”
The 30-year-old added: “This has been the most incredible experience of my life, I’ve learned to dance, I’ve made a best friend, I’ve been on the show that I loved for 12 years.”
An emotional Clifton said she “can’t believe it” and told Oduba “you’re a dancer” before the pair were lofted in the air by the show’s professional dancers and celebrities.
Each duo performed three separate routines in the final, which marked the last time head judge Len Goodman would preside over the dance floor.
Oduba, watched by his wife and parents in the audience, collected his first perfect 40 of the series in the final for a show dance to jazz hit I Got Rhythm, which was praised by Goodman, who said he “couldn’t have asked for anything more”.
They also scored 40 for their jive to Bruno Mars’s hit Runaway Baby from week four, which Craig Revel Horwood called “virtuosic” and Bruno Tonioli said was “sheer perfection”.
Their first dance was a take on the Gene Kelly classic I’m Singin’ In The Rain, which was watched by Kelly’s widow, Patricia Ward, after she had praised the routine when the couple first performed it in week three.
The routine, an American smooth, was lauded as “heaven to watch” whilst Oduba said having Ward there was “hard to top”.
This year’s competition marks the show’s most-watched series in its 12-year history, with an average of 11.3 million viewers tuning in to BBC One’s flagship entertainment show on Saturday nights.
During the final, the show’s production team and audience had given a standing ovation to Goodman, who has starred as head judge since the programme began in 2004.
A number of the show’s stars past and present commented on his departure in a video montage, including 2014 winner Caroline Flack, who called him the “perfect gentleman”, and professional dancer Anton du Beke, who said Goodman “brings joy to so many people”.
Goodman, 72, said: “I’ll miss just turning up and the anticipation of what is going to occur, I will miss what I consider the three best judges on television, being close to the band and the fabulous singers.
“I think the whole thing is like a big machine with cogs. Every cog has to work to make the whole thing happen. You’d miss any one of the cogs because that’s what makes Strictly into Strictly.
“I’m going to miss it all,” he added, before the programme’s professional dancers performed a classic ballroom dance as a tribute.
Redknapp and Kevin Clifton - who has now lost four Strictly finals in a row - brought Tonioli to tears earlier in the show with a moving show dance to One Moment In Time, which told the story of the former Eternal singer’s Strictly journey.
They scored 38 for the routine and were also awarded 38 from the judges for their cha cha to Flashdance ... What A Feeling, which Goodman called a “red hot cha cha from a red hot Redknapp”.
Their final dance, an Argentine tango to Tanguera by Sexteto Mayor, handed them the perfect 40 points from the judges, whose scores only act as guidance in the final.
Favourite for most of the competition, Mac, got off to a mixed start as he was marked down during his first routine.
The ex-Hollyoaks actor and Mabuse scored 36 for their quickstep from week four to Frank Sinatra’s I Won’t Dance but matched Oduba and Clifton’s 40 with an interpretive show dance to Adele’s Set Fire To The Rain.
Their record-breaking routine to Magalenha by Sergio Mendes again scored 40 points after it became the first ever samba to be given full marks in the show’s history when the pair performed it for the first time in week 10.