Early touchdowns from Burrow and Tee Higgins lifted Cincinnati to a 14-10 half-time lead, before Patrick Mahomes and the Chiefs exploded for a 14-point third quarter.
A field goal at the start of the fourth brought the Bengals back within four, and they regained the lead with eight minutes to go when Burrow found Chris Evans in the end zone.
Cincinnati's defence then held strong down the stretch and the Bengals were ultimately able to secure the three-point win.
It was a similar result in Atlanta, with the Pittsburgh Steelers edging past the Falcons 19-16.
The difference came in the dying seconds when Falcons quarterback Marcus Mariota was intercepted by Minkah Fitzpatrick.
The result marks Mariota's fourth loss in five starts, while the Steelers have won back-to-back games for the first time this season.
Aaron Rodgers and the Green Bay Packers fought back from an early deficit to eclipse the Chicago Bears 28-19.
Chicago quarterback Justin Fields had a monster game in every statistic but the one that matters most, finishing with 71 yards rushing and 254 yards passing, but no touchdowns.
Overtime was not enough to separate the Washington Commanders and New York Giants, with both teams grinding out a 20-20 draw.
The Giants had the chance to win it at the death, but kicker Graham Gano's 58-yard attempt fell well short.
Elsewhere, the Detroit Lions had their best showing of the year in a 40-14 demolition of the Jacksonville Jaguars.
The Miami Dolphins had their five-game winning streak snapped by the San Francisco 49ers in a 33-17 defeat.
A late two-yard rushing touchdown from second-string quarterback Tyler Huntley pushed the Baltimore Ravens past the Denver Broncos 10-9 in a dour affair.
Jalen Hurts threw for 380 yards and three touchdowns as the Philadelphia Eagles cruised past the Tennessee Titans 35-10.
The Dallas Cowboys, Minnesota Vikings, Cleveland Browns, Seattle Seahawks and Las Vegas Raiders rounded out Sunday's victors.