Ronaldo sent Tim Krul the wrong way on 75 minutes after the Portuguese had been hauled down by Max Aarons on the edge of the six-yard box.
The Red Devils had offered little attacking threat before then, though Alex Telles hit the bar with a deflected free-kick during the first half.
They also had David de Gea to thank for two fine stops to deny Teemu Pukki and Ozan Kabak as Norwich put in a spirited display.
There was concern for Victor Lindelof when he was forced off during the second half holding his chest, though Rangnick said the Sweden defender had had tests and “seems OK”.
The win moves Manchester United above Arsenal and Tottenham into fifth place. Norwich stay bottom on goal difference, three points from safety.
Ronaldo rescues United
After making made a raft of changes for the midweek Champions League draw with Young Boys, Ralf Rangnick restored the XI that beat Crystal Palace in his first game in charge last weekend.
But despite having a full week to prepare, United looked sluggish and rarely threatened against a side who have conceded more goals than any other in the Premier League this season.
Ronaldo had to wait until the 38th minute to get his first sight of goal, running on to Scott McTominay’s pass before cutting inside and curling in a left-foot shot that was palmed away by Krul.
United faded further in the second half but, like against Palace, managed to find a way, with Ronaldo emphatically dispatching the match-winning spot-kick after Aarons had dragged him down while attempting to meet a right-wing cross.
De Gea also maintained his fine form this season, diving full stretch to tip a fizzing Pukki drive over the bar before clawing away Kabak’s goalbound header late on.
It means the Red Devils have kept successive Premier League clean sheets for the first time since a run of four from February to March 2021, and have won three consecutive league games for only the second time this season.
There is also a bit of history for Rangnick, who became the second Manchester United manager to register a clean sheet in his first two league games in charge after Ernest Mangnall in 1903.
Spirited Norwich fall just short
Injury, ineligibility and illness forced Norwich boss Dean Smith into five changes to the side that lost at Tottenham last week, including a first start of the season for Przemyslaw Plachet, and recalls for Kabak and Dimitrios Giannoulis – whose last start was the 7-0 defeat at Chelsea in October.
They suffered a further injury blow after 20 minutes when captain Grant Hanley was forced off with a shoulder injury, with midfielder Jacob Sorensen having to come on and play alongside Kabak in the Canaries’ defence.
But despite the patched-up nature of the side it was a performance full of effort and endeavour.
A lack of attacking threat again let them down – no team has scored fewer than Norwich’s eight goals in the Premier League this season – though they can rightly point to an impressive performance by De Gea.
There was also some good news on the injury front when Todd Cantwell came off the bench in the second half for just his second appearance since September.
Despite the disappointment of defeat it was a performance to build on as Smith prepares to welcome former club Aston Villa to Carrow Road next week.