United gave Solskjaer a three-year contract last summer believing the 49-year-old was on course to bring the glory days back.
However, he was then axed in November after struggling to get the best from his star-studded squad.
Links to Mauricio Pochettino and Ajax’s Erik ten Hag then arose but United opted to appoint an interim boss instead.
And, after holding talks with the likes of Lucien Favre, Rui Garcia and Ernesto Valverde, it was Rangnick who was given the job until the end of the season.
Since his appointment, the club have lost just one of their last 13 Premier League matches.
That was against Wolves in January, though they have exited the FA Cup at the hands of Middlesbrough as well.
And Rangnick believes that shows his side are better than when Solskjaer ruled the roost at Old Trafford.
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Speaking ahead of Sunday’s match with City, he said: “When City won 2-0 at Old Trafford they did it in style and it could have been a higher margin.
“Now, for us, it is about confirming the development of the team in the last three months.
“We have been unbeaten in the last 11 games and we haven’t lost a single away game since I arrived.
“We know this will be the biggest challenge so far because we haven’t played against a team of that kind of quality.”
On his tactics for the match at the Etihad, he said: “We have to be as compact as we can be and play as much as a team as we can.
“It will be challenging for us. We will have moments in the game when we have to chase the ball and be without it for one or two minutes.
“But this shouldn’t happen all the time. It’s about being on the front foot, being courageous, and creating moments when we can put them under pressure.
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“We need to perform at the highest possible level – and I have no doubt we have players who can do that. But obviously it’s about showing it on the pitch.”
Recently, there have been suggestions of a divide within United.
Reports have suggested squad members are unhappy with skipper Harry Maguire, who has struggled for consistency this term.
Yet Rangnick has insisted they all get along at Old Trafford and that they have ‘full focus’ in every training session.
“Players should know they have to perform to play or even for a new contract or to be kept in the club,” said the 63-year-old.
“That should be the case all the time in professional football.
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“That might be a bit simplistic, but for us it’s about getting the best out of this current squad.
“I can assure you the atmosphere in the locker room, the focus in training, we have no issues.
“All that about outside noises, about players not being focused in training, is not correct.
“They get on well with each other, they have a full focus in every session.
“It’s all about getting the best possible result in the coming games.”
United beat City 2-0 in the same fixture last season.
Bruno Fernandes opened the scoring via the penalty spot, with Luke Shaw then doubling their lead much to Solskjaer’s delight.
Solskjaer, for his part, said after his sacking in November he was ‘proud’ of his United legacy.
“Very, very proud,” he said.
“Of course, it’s one of those things you dream of in your life. When you’ve been a player, when you’ve been a reserve team coach, the next job then, the only dream and the only thing you haven’t done is to manage the club, and I have now.
“It’s been a ball, it’s been absolutely enjoyable from the first to the last minute, so I have to thank all the players because since I came in, they’re top lads, top people.
“Some have come, some have gone, but all of them, they’ve been such an enjoyable bunch to work with.”
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