Sir Bradley, Ed Clancy, Steven Burke and Owain Doull beat Australia in a nail-biting finish in a head-to-head ride off against Australia.
It means Sir Bradley is now the most decorated British Olympic athlete of all time with eight medals including five golds, while Clancy is now a three-time Olympic champion.
The team finished the 4km race in three minutes, 50.265 seconds.
It was Great Britain’s third straight Olympic title in the event. Denmark took the bronze medal ahead of New Zealand.
The quartet’s win sees Team GB rise to third in the overall medals table, leapfrogging Japan.
After receiving a hug from fellow Olympic knights Sir Steve Redgrave and Sir Chris Hoy - having surpassed the latter’s British medal record - Sir Bradley praised his teammates, saying: “The last 12 months we have pretty much done everything together, training camps at altitude, early-morning starts at the track, late finishes at the track before Christmas Day and all for this.
“We are here and we have done it and it is just this four guys here.
“I would never have come back if we didn’t have the calibre - I have always said that Ed and Burkey for me are two of the most underrated athletes I have ever raced with, so talented at what they do and they don’t get the credit for it because they are not the big road stars. (And) you have someone like Doully here who ... can do anything in the sport and nothing fazes him, especially Olympic finals at 21.
When you are with guys like that on the line it makes your job a hell of a lot easier. In some ways I realised what we were going into and that adds nerves to it, these guys (were) bouncing of the ceiling all afternoon in the apartment.”
Clancy said that his third gold medal was “the best of them all”.
He said: “Truth be told, we haven’t won a fat deal between the London Olympics and now, and we have had some big downs.
“Crossing the line a second ahead of the Aussies made every single pedal rev and every training session worthwhile.”
The 31-year-old Yorkshireman also praised medical staff who helped him overcome a recent back injury, saying: “I can’t thank them enough.”
Clancy’s proud family were “in bits” as they watched him win gold at their home in Tetbury, Gloucestershire.
His brother Alex, 32, told the Press Association: “It is phenomenal. We are so proud of him. What an amazing achievement. The family has been in tears. Just eight month ago he had a prolapsed disc. He could not walk, let alone ride a bike and he had to go through a lot of rehabilitation.
“Day to day he came through it and to go from that to win in such an amazing race that was so close - has left us speechless.
“We are really really proud. We are all just in bits.
“We were all in tears watching it on TV because it was such a tight race.
“His last lap was like superhero strength. I honestly do not know where he came from.”
Burke’s father Alvin, 53, said the family were “over the moon” after his son won his second Olympic gold medal.
He and Steven’s mother Sharon were joined by their extended family to watch at their home in Colne, Lancashire.
Alvin told the Press Association: “That was brilliant. How close was that race? We are really, really, over the moon.
“It was the closest race I have ever seen. It was brilliant. I think he did really well.
“He has put so much effort in, and to get a world record as well. He did the business.”
He added that he hoped his son, now 28, would carry on and take part in the 2020 games in Tokyo.