City re-established a six-point lead at the top of the table over Liverpool and opened up a 22-point advantage over United, whose chances of just qualifying for the Champions League next season are fading fast.
Rangnick’s men now trail Arsenal, who also have three games in hand, by a point in fourth.
“It is a difficult game that shows we have a long way to go to close that gap,” said the German.
“Everyone knows how good they are. They are one of the top teams in the world and there is a gap between the two teams.”
At the top of the table, Liverpool also have a game in hand to reduce City’s advantage to three points.
But Pep Guardiola’s men responded to the pressure applied by the Reds’ seven-game winning run in the league by bouncing back from two sub-par performances against Tottenham and Everton.
Just as in their meeting at Old Trafford earlier in the season, City cruised to victory as they secured just a second league double over their local rivals since an Abu Dhabi takeover in 2008 swung the balance of power in Manchester.
Former United captain Roy Keane accused some United players of “giving up” and claimed they should not play for the club again.
And Guardiola summed up the difference in desire as well as quality between the teams.
“The biggest success for me as a manager is that after three Premier League titles is that still we run like we run, and are there to try (to win the league) again,” he said.
United were without the injured Cristiano Ronaldo and Edinson Cavani in attack, but it was defenders Raphael Varane and Luke Shaw, missing due to positive cases of coronavirus, whose presence was most keenly missed.
The visitors were breached by City’s first sustained attack as Bernardo Silva’s low cross found De Bruyne completely unmarked to sweep home from just inside the box.
Despite their dominance over United when it comes to finishing higher in the table in recent years, City had only won one of the previous seven derbies at the Etihad.
United have consistently had joy on the counter-attack in that run and the pace of Jadon Sancho and Anthony Elanga caused problems once more.
Sancho left City as a 17-year-old to make his name at Borussia Dortmund after growing tired of waiting for a pathway into the first team.
The England international took time to find his feet at United after returning to the Premier League for huge £73 million fee.
But Sancho has been one of the few successes of Rangnick’s spell in charge and produced a stunning finish into the far corner on 22 minutes.
United captain Harry Maguire had to cope with more reports this week of members of his own team questioning whether he is of the required standard to lead the team.
More criticism is sure to come Maguire’s way after another leaden-footed display summed up by City’s second goal.
Phil Foden looped the ball over his England teammate and when he was denied by a brilliant save from David de Gea, the loose ball finally found its way to De Bruyne to smash home.
Rangnick’s men then had to sustain a period of pressure just to get to half-time still in the game.
Foden fired wide from a perfect De Bruyne pass before more David De Gea heroics prevented Mahrez finding the bottom corner.
The one way traffic continued after the break and only another day this season when De Gea was United’s best performer prevented an even more embarrassing rout.
Mahrez made the game safe 22 minutes from time when he was picked out by De Bruyne’s corner and connected sweetly to find the far corner.
The Algerian then rubbed salt into United wounds in stoppage time after VAR ruled the City winger was not offside when he blasted high past De Gea.
Arsenal beat Watford to climb into top four
In the earlier game yesterday, Bukayo Saka starred as Arsenal leapfrogged Manchester United into the Premier League top four with a 3-2 win at struggling Watford.
Martin Odegaard struck early for the visitors before Cucho Hernandez equalised with a stunning overhead kick minutes later but England star Saka restored Arsenal’s lead in an entertaining first half.
Brazilian winger Gabriel Martinelli made it 3-1 early in the second half and the Gunners held out for a crucial win despite a late goal from Moussa Sissoko.
The victory lifted Mikel Arteta’s men into fourth place in the Premier League - one point ahead of United, who were in action in the later kick-off against leaders Manchester City.
Arteta, whose team started the season with three straight league defeats, said it was “very pleasing” to be in the top four, although he felt the win should have been more comfortable.
“We know how difficult it is to be in this position in this league,” he told Sky Sports. “We’ve earned it - we have come a long way as a team.”
Arteta was asked about the role he played in the build-up to Arsenal’s third goal - he grabbed the ball when it went out of play and quickly passed it to Saka, who launched the ultimately decisive move with a throw-in.
Arsenal jokingly awarded their manager an assist on their Twitter feed.
“I saw an advantage and I took it,” said Arteta. “We have to be alive and that is the way I live the game. I try to instil that in the team as well.”
Watford had the ball in the net a few seconds after kick-off but Emmanuel Dennis’s effort was ruled out for offside.
Arsenal settled quickly after their early fright and were in front in the fifth minute after neat work down the right, with Saka cutting back to Odegaard, who finished with his left foot.
But the Gunners’ advantage was cancelled out in the 11th minute when Colombian forward Hernandez produced an acrobatic overhead kick from near the penalty spot after a cross from Kiko Femenia.
Arsenal restored their advantage just after the half-hour mark with a cleverly crafted goal finished by the lively Saka.
The England man dispossessed Tom Cleverley just outside the Watford box before passing to Alexandre Lacazette, who held the ball up then backheeled it to Saka, who fired in.
Arsenal extended their lead early in the second half with a delightful third from Martinelli, who hammered home from the edge of the box after a neat layoff from Lacazette.
Watford still offered a threat despite Arsenal’s superiority and Dennis should have narrowed the gap but miscued a chance at the far post with about 20 minutes to go.
Arsenal substitute Eddie Nketiah hit the woodwork with 10 minutes left before Sissoko gave the visitors a late scare when he bundled the ball home in the 87th minute.
The win makes Arsenal strong favourites to finish in the top four and qualify for next season’s Champions League - a competition they have not played in since the 2016/17 season.
They have 48 points after 25 games, one ahead of United, who had played two games more ahead of the Manchester derby.
Roy Hodgson’s Watford remain second from bottom of the Premier League table, three points from safety.