The match began with a touchdown from star receiver Odell Beckham Jr, swaying in LA's favour.
Stafford -who arrived in Los Angeles a year ago after spending 11 years at Detroit- was able to control the match up until half time.
A highlight of the event, the half-time entertainment featured an all-star cast of Dr Dre, Eminem, Snoop Dogg and 50 Cent.
The home squad faced a challenge of their own, when Beckham injured his knee and was forced to sit the rest of the game.
25-year-old Burrow made an instant impact in the second half, navigating the championship-less Bengals to a narrow lead entering the fourth quarter.
It was the home crowd who aided the Rams' last few drives, as Cooper Kupp stepped up in the absence of Beckham to score a touchdown with minutes remaining.
Kupp was selected as the Super Bowl MVP as coach Sean McVay hoisted the prestigious trophy - capping off another season of American football.
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