Johnson, 38, rode Menorah to victory in the Oaksey Chase at Sandown, his 235th win of the season. Closest rival Aidan Coleman finished second on 130 wins.
"It's always been my dream," said Johnson. "There's no feeling like it."
Paul Nicholls won the trainers' title - his 10th in 11 seasons - ahead of Willie Mullins.
Read more: Champion at last - Johnson, the 16-time runner-up
McCoy was champion jockey for 20 years in a row from 1996 to 2015, with Johnson runner-up 16 times.
Johnson was given a guard of honour by his fellow jockeys on his way to being handed the champion's trophy by McCoy.
"I've waited 20 years for this day and I'm going to enjoy it," said Johnson.
McCoy retired at this meeting last year, having ridden 231 winners in the season.
Johnson admitted McCoy's retirement brought a "pressure" to win the title, but his tally of 235 winners is the highest since McCoy rode 258 in 2003.
He was handed a new trophy, engraved with an image of McCoy winning the 2010 Grand National and the words "Bravery, Dedication, Resilience, Talent".
AP McCoy: "We sat together for 20 years. It's a big day and I know I wouldn't have achieved what I did without him driving me on up and down the country. For me, he has always been a champion. He thoroughly deserves it."
Trainer David Pipe: "Day in, day out, he will give 110% and that's what's called a champion jockey."
Jockey Aidan Coleman: "He's been an unbelievable ambassador for countless years. He's getting his rewards and deserves it. Unfortunately for us other lads, I think he will be getting a few more."
Cornelius Lysaght, BBC horse racing correspondent:
"Richard Johnson told 5 Live Sport that he was looking forward to trying to 'retain the title rather than chase it' as he did when runner-up to AP McCoy 16 times. And he must have an exceptional chance of a repeat.
"Ok, everyone knows about the perils associated with jump racing, but injury permitting Johnson's won this by a huge, positively McCoy-esque margin of over 100 successes, and trainer John Ferguson who provided heading for half the wins of his closest pursuer, Aidan Coleman, has packed up. It all bodes well."
While Johnson's cushion to lift the jockeys' title was comfortable, the trainers' championship was not decided until Sandown's Bet365 Gold Cup Chase, the feature race of the final meeting of the season.
The Young Master - ridden by Sam Waley-Cohen and trained by Neil Mulholland - claimed the Gold Cup but the subsequent places decided the trainers' title.
Nicholls, who brought 19 horses to the Surrey course, had Just a Par and Southfield Theatre placed second and fourth, giving him an unassailable prize money total for the season.
Mullins had taken 10 charges to Sandown in a bid to win the title for the first time but Ireland's wait for a champion trainer goes on, with Vincent O'Brien the last to succeed in 1954.
Cornelius Lysaght, BBC horse racing correspondent:
"With just two successes in valuable Grade One races - compared to 13 for Willie Mullins - Paul Nicholls has really done tremendously well to hold on to his title.
"Now Mullins is speaking of his determination to try to win the next campaign, in which case it could be harder than ever for Nicholls because this time around the County Carlow trainer only started concentrating on the British title last month.
"But next season, expect the ferries to be buzzing with Mullins' runners as soon as the core jumps season starts in late October."
Earlier in the day, Nicky-Henderson-trained Sprinter Sacre recorded his fourth straight win, landing the Celebration Chase by 15 lengths.
The 10-year-old, who was diagnosed with an irregular heartbeat in 2014, went off as 11-10 favourite and under the ride of Nico de Boinville showed his class to further underline his rejuvenation under Henderson.
"That was why we had to run. He loves doing this - if you look at that you can see why we have to do it," said Henderson.
"There were moments we thought it was all over but we were never under any pressure.
"It's been a long journey but this season has been amazing. He's great for the racing game. I hope he's good for me but my old heart might not take much more of it."
BBC horse racing correspondent Cornelius Lysaght said: "This has been amongst the most enjoyable national hunt seasons ever, not least because of the rejuvenation of Sprinter Sacre.
"His return from jumping's wilderness to win four from four has been nothing short of extraordinary.
"I'm not sure those directly around him can quite believe it either, but they, especially his trainer Nicky Henderson and team, who did the nursing, deserve huge credit and a big thank you - this horse isn't admired, he's adored."
Richard Johnson has become the first man to be crowned jump racing's champion jockey after the AP McCoy era, ending a 20-year wait for the title.