Trevoh Chalobah and Mason Mount came through with the early goals for Chelsea before Norwich threatened a comeback when Teemu Pukki found the net with a second-half penalty. Kai Havertz, though, settled Chelsea's nerves with a thumping 90th-minute finish.
The result will come as a much-needed relief for the London club. Just hours before kickoff, Chelsea were thrown into chaos when owner Roman Abramovich was sanctioned by the U.K. government for his ties with Russia President Vladimir Putin. The shock announcement means Abramovich's attempts to sell the club are on hold for the time being.
The move will also have a profound impact on the day-to-day running of the club. While they can continue to fulfill their fixtures, they can no longer sell new tickets for future matches and the club shop will have to close. They also cannot agree new contracts for existing players, a particularly urgent situation in the case of Andreas Christensen, Cesar Azpilicueta and Antonio Rudiger whose existing deals all expire at the end of the season. In addition, they are prohibited from agreeing to any new transactions for the summer transfer window.
Tuchel admitted it had been a testing day but was impressed with how his squad responded once the game got underway.
"[It] Would be a lie if I said I had no doubts [about how team would perform], but I also had trust because in difficult circumstances, we've produced a lot of performances, a lot of results recently," he said on NBC Sports after the final whistle. "So we can trust in our mentality and the culture in the club, and again we prove that we allow ourselves to focus on football and sports, and we are people who work in sports, nothing else.
"It's not been as difficult as people may think," the Chelsea boss added. "The noise is there, it's of course another level of distraction, but still the team meeting was the same, the schedule going to [Norwich] was the same, but of course the subject is there, the talks are going on.
"There's a certain distraction because of the level of impact of today's news is big, but at the same time, we don't know how big. What does it help [to focus on it]? We cannot influence it, it's a fact and that's why in the end we said let's focus and let's try to enjoy because we love the game, we love the sport, we love the competition and we love the Premier League."
With signings currently prohibited, the two early goals offered a particular cause for optimism for Chelsea's travelling fans, who sang Abramovich's name loudly before kickoff and frequently during the 90 minutes. Two graduates of the club's academy, a pipeline that could become increasingly important amid the uncertainty over Chelsea's ownership, fired the visitors into a 2-0 lead inside 14 minutes.
In just the third minute, centre-back Chalobah was allowed to rise highest at the near post to head home a corner from Mount. It wasn't long before Mount himself found the net. After some neat footwork and a clever layoff from Havertz, playing as a false nine, Mount took a delightful first touch and then fired an unstoppable shot into the net beyond Tim Krul.
In truth, after such a tumultuous few hours and days, Chelsea could hardly have wished for better opponents than a Norwich team that had lost their last five Premier League games to leave them rooted to the bottom of the table.
Still wearing the logo of sponsor Three on their shirts, despite the telecommunications company announcing shortly before kickoff that they had decided to suspend their deal with the club, Chelsea controlled proceedings throughout the opening 45 minutes. Had they been more ruthless in front of goal, Chelsea could have put the game out of reach by the interval.
Instead, boosted by a double substitution at half-time that saw the introduction of Lukas Rupp and Milot Rashica, Norwich made the game a true contest in the second half. And they got their reward when Chalobah's evening took a turn for the worse as he was penalised for a hand-ball in the box following a VAR review. Pukki sent Edouard Mendy the wrong way from the penalty spot to ignite the home crowd for the first time.
Norwich continued to threaten but Havertz finally settled the result as the clock ticked toward stoppage time to avoid another blow for Chelsea on an unprecedented day for the club.
The result keeps Chelsea comfortably in third place, above the scramble for fourth spot. A Champions League berth for next season continues to look a formality. Much of the rest of the club's future, though, is now shrouded in doubt.