Appearing in its 15th FA Cup quarter-final since the turn of the century – the most of any side – Chelsea was not at its sharpest at Stamford Bridge but, as it has done in many of its 14 matches under Tuchel so far, managed to grind out a result.
Norwood's own goal put Chelsea ahead, with David McGoldrick missing a golden chance to restore parity when he headed wide from five yards out.
Chelsea made its fortune count, holding firm in rather nervy fashion before hitting the Blades on the break and securing its progression thanks to Ziyech, reaching the semi-finals for the fourth time in five seasons.
Playing at the end of a difficult week personally following the death of his brother, Phil Jagielka might have put United ahead early on but failed to generate enough power on his header to trouble Kepa Arrizabalaga.
Yet United's bright start ultimately proved fruitless – Chelsea striking first when Norwood diverted Ben Chilwell's effort beyond Aaron Ramsdale.
Norwood was almost at fault for another goal before half-time when his stray pass was intercepted by Christian Pulisic, though Ramsdale rushed out to spare his team-mate's blushes.
Displaying some terrific footwork, Pulisic engineered another chance straight after the restart, but Ramsdale again stood firm.
McGoldrick should have built on Ramsdale's save, only to head wide of a gaping goal from point-blank range.
Kepa had work to do soon after – McGoldrick's strike partner Oli McBurnie forcing a smart save out of the Spaniard.
Rhian Brewster then saw a strike deflected into the side netting, yet for all United's pressure, Chelsea dealt the knockout blow when substitute Ziyech drilled in from close range at the end of a sweeping counter to ensure the Blues progressed.