Collins picked up a wayward pass from the Rams' Case Keenum before weaving past eight would-be tacklers for a memorable touchdown as he zig-zagged across the field.
"It's my first career touchdown in the NFL. I'm never going to forget it. Definitely a game-changing moment for our team," he said.
"I headed to one sideline, looked up field and saw a lot of blue. I glanced to my left and saw a lot of white so I went over there to my guys and they led me through to the end zone."
The stunning score had the Giants level at 10-10 at the half, the Rams having led 10-0 off the back of a Tavon Austin touchdown and a field goal.
The Giants' star receiver Odell Beckham Jr was the focus of the pre-match attention and played his part in the opening Robbie Gould field goal to put his side on the scoreboard.
Then came the moment of magic for Collins, to light up the 16th NFL game played in London, the others at Wembley Stadium.
Collins continued his heroics in the fourth quarter, helping the Giants into Rams' territory with another decisive run.
A final drive saw Rashad Jennings go over from one-yard to give the New York franchise the lead for the first time and to keep its playoff hopes alive.
The Rams pushed hard to level, but both drives late in the fourth quarter ended with interceptions, the final one with only 50 seconds left by Dominique Rodgers-Cromartie.
It secured victory for the Giants -- now 4-3.
Twickenham was hosting a sporting event other than rugby union for the first time in its 107-year history as the NFL continues its expansion plans in the UK capital. It is the second time this month the famous ground has set a first, with the Argentina Pumas hosting a match in the Southern Hemisphere Rugby Championship.
Sunday's NFL clash was the second of a three-match series in London this month, with the other two hosted by Wembley.
The Jacksonville Jaguars beat the Indianapolis Colts 30-27 in the first, and the Washington Redskins face the Cincinnati Bengals in the third next Sunday.