The Bengals went down 23-20 to the Steelers after missing kicks at the end of regulation and in the additional period that would have secured a winning start.
Instead, Chris Boswell kicked the game-winning field goal for the Steelers in the final seconds of overtime.
Burrow had put the Bengals in position to succeed despite his dismal start, looking nothing like the 2021 Comeback Player of the Year who led Cincinnati to the Super Bowl.
The quarterback had four turnovers before half-time – a career high for an entire game – including a pick-six for Minkah Fitzpatrick from his very first pass.
Burrow finished with four interceptions, a lost fumble and seven sacks, but he also threw two touchdown passes, including one for Ja'Marr Chase that appeared to set up a stunning Bengals win.
However, Fitzpatrick blocked Evan McPherson's PAT, and McPherson missed his target again in overtime.
Boswell similarly squandered a chance to win the game, hitting the left post, but he made amends just as it looked as though the NFL would see two ties in the same week for the first time since Week 2 in 1973, with the Houston Texans and Indianapolis Colts also tied 20-20.
Browns deny Baker's Panthers
The Bengals were the only team not to win in the AFC North – a division that had it all on Sunday. While Lamar Jackson guided the Baltimore Ravens to victory over the New York Jets, the Cleveland Browns celebrated their first Week 1 win since 2004 as former QB Baker Mayfield saw his Carolina Panthers beaten 24-23 by a late field goal.
The Miami Dolphins' 20-7 win over AFC East rivals the Patriots was far more straightforward, with Tua Tagovailoa becoming the first QB to remain unbeaten against Bill Belichick in the regular season having started at least four games against New England.
49ers flounder at sodden Soldier Field
Miserable conditions in Chicago hampered the San Francisco 49ers as they suffered a shock 19-10 loss to the Bears, with Trey Lance's miserable 13-of-28 passing including an awful fourth-quarter interception for a quarterback rating of just 50.3.
Elsewhere in the NFC, Jalen Hurts also did not throw a single touchdown pass in the Philadelphia Eagles' 38-35 victory over the Detroit Lions – he contributed one of four rushing scores – but the QB connected with A.J. Brown for 155 yards on 10 catches. Marquee offseason trade signing Brown had 128 receiving yards before half-time, the most in the first half of a team debut by any non-rookie since at least 1991.