On the final sequence, Patriots LB Kyle Van Noy missed a tackle on Drake, and Rob Gronkowski - who was deep as a safety - and J.C. Jackson couldn't take down Drake as the streaked into the end zone to cap the mayhem in Miami.
In what has already been dubbed "The Miami Miracle", the Dolphins pulled out a 34-33 victory. "When you practice it, you don't really know how it's going to turn out". He outran several Patriots, and the final man between him and the end zone was Gronkowski.
Belichick opted to have Gostkowski kick a 22-yarder in the closing seconds rather than pin the Dolphins near their goal line.
The Dolphins (7-6) prevented the Patriots (9-4) from clinching the AFC East for a 10th consecutive time on their home field. "We work on it every Friday and Saturday", Gase said.
Dolphins take down the Patriots on last-second hook and ladder
"We had them right where we wanted", Tannehill said dryly.
Miami's victory moved them to 7-6 while the Patriots slipped to 9-4 in a game that also witnessed yet another landmark for Tom Brady, who moved clear of Peyton Manning with his 580th career touchdown pass (including playoffs) - a new National Football League record.
Gronkowski's postgame reaction on the loss: "It sucked".
The Chiefs defence forced a turnover but Harrison Butker missed from 43 yards with a potential game-winning field goal.
The Dolphins started the play with seven seconds left in the game and on their own 31 yard line.
The miracle play will have a huge impact on the playoff picture.