So much for "control".
Villa were unlucky not to peg United back after Scott McTominay's early - and, in fairness, excellent - header that gave Rangnick's side the lead; the visitors had two balls in the net that were overturned for obstruction and offside, respectively.
But warning signs still remain at Old Trafford, with a host of individuals failing to show the standards required of a Manchester United side.
United disjointed again
Passing was once again a key issue for the Red Devils against Villa; with the midfield pairing of McTominay and Fred seemingly evaporating as the minutes ticked away.
The two central midfielders combined for a neat goal that gave the hosts an eighth-minute lead, but there was a lack of tenacity in the middle as the match progressed.
United are still desperate for a holding midfielder who can break up attacks, recycle possession as well as being able to find a decent pass; their drop-off in the second period was an indictment of the lack of care that area of the pitch has been given over the last three years.
While striker Edinson Cavani posed a dangerous threat to the Villa backline throughout the evening's play, he received little creativity from the wings, with neither Mason Greenwood nor Marcus Rashford stamping an impact on the clash.
Rashford, in particular, had a very curious performance on Sir Matt Busby Way.
The inside-forward has seemed at odds with Rangnick's 4-2-2-2 system, but was ineffective when returned to a more conventional left-wing role, with his movement often cut off by Matty Cash and the stoic Villa defence.
Rashford has struggled since his return to the side in the Autumn, and another poor showing on Monday evening may well put his starting place in jeopardy, with the likes of Jesse Lingard and Anthony Elanga pushing for more playing time.
The 24-year-old, when in more central positions, didn't convince either - a loose ball from a Greenwood effort could have been capitalised on, but Rashford refused to make a play for the ball that could have been chased.
Indeed, Lingard and Elanga looked fairly convincing when introduced from the bench in the late stages, the Swedish attacker especially provided an all-too-rare moment of excitement by actually taking on his defender and beating him on the outside.
The Old Trafford faithful rose to their feet in delight, but there were no attackers willing to make an attacking run to stab the ball home to wrap up the victory.
While nothing should be taken away from Villa, who on another night would be looking ahead to their Fourth Round tie, United managed to secure their place in the next round where they will host EFL Championship side Middlesbrough.
Rangnick still has work to do
Everybody knew this appointment would take time. Ralf Rangnick is a proven football manager, but if there were any illusions over the quality of his players, they will have been shattered now.
The road ahead is a long one for the German, who has agreed a two-year consultancy role at Old Trafford once his interim spell concludes in May.
Perhaps the most confusing question that remains, however, is where the effort has gone. There hasn't been a 'new manager bounce' as some may have expected from the players, but the final message should not be confused - Rangnick is not going anywhere anytime soon.
The German still retains full support from Football Director John Murtough. Additionally, a big factor is how to define the 'consultancy' role from the end of the year - but it begs the question that surely Rangnick isn't going to leave a well-paid job running Lokomotiv Moscow's sports department to have little influence at Manchester United?
The club isn't going to abandon ship here - United have been largely wise off the pitch since Solskjaer's appointment three years ago, but there is still much to do if United are to compete with the best clubs - not only in England but in Europe.
However, if there is one thing to be learned from Monday evening's game, it's that they are currently nowhere near.