Chelsea remain undefeated in eight games under interim manager Guus Hiddink and an impressive fourth-round display, albeit against second-tier opposition, confirmed the progress that they have made under the Dutchman.
The win means that Chelsea remain in contention to repeat the FA Cup success they enjoyed when Hiddink was last in interim charge in 2009, but they were handed a testing draw at home to City.
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Speaking before the draw was made, Hiddink told the BBC: "We are delighted. Because if you play this game, it's always difficult.
"If you play the big games, the big teams, the concentration is always high automatically, but these games you have to be well prepared mentally.
"We played well in the first half, but we weren't finishing it off in the first 20 minutes. Then they got the equaliser, but we reacted very well."
Oscar's first-half treble grabbed the attention at Stadium MK, but Hazard's penalty, his first goal in 30 appearances, came as a huge relief, while substitute Bertrand Traore netted his first Chelsea goal.
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Hiddink signalled his intentions by fielding a strong side, although he did take the opportunity to hand rare starts to left-back Baba Rahman and young midfielder Ruben Loftus-Cheek.
With loan signing Alexandre Pato not eligible to play and both Loic Remy and Radamel Falcao injured, Diego Costa led the line and his work alongside Oscar quickly nixed any chance of an upset.
After a dominant start, Chelsea took a 15th-minute lead courtesy of a defensive mistake.
An underhit back-pass by Kyle McFadzean allowed Costa to advance on goal and the striker unselfishly squared the ball for Oscar to finish into an empty net.
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It was to MK Dons' credit that they responded positively and drew level six minutes later when Darren Potter's shot took a deflection off Nemanja Matic and looped over the leaping Thibaut Courtois.
However, the gulf in quality between the teams meant it was only a matter of time before Chelsea asserted their superiority, although goalkeeper David Martin did his best to hold them at bay, saving well from Hazard before Costa nicked the ball off the toe of Oscar in front of goal.
Another chance quickly came Oscar's way and this time the Brazilian made no mistake with a well-placed finish in the 32nd minute.
The playmaker, operating on the right on this occasion with Loftus-Cheek filling the central role, completed his hat-trick when he cut in from the left and found the bottom-right corner from outside the box.
The home side had offered little resistance and the pattern of the game was maintained after the break when Hazard was tripped by Potter in the 55th minute.
Chelsea skipper John Terry intervened to ensure Hazard took the penalty and the Belgium international rolled the ball home to his obvious relief.
Substitute Traore then completed a routine victory for the visitors when he finished with a first-time shot after Hazard had chased Cesc Fabregas's pass out to the byline before setting up the Burkina Faso youngster.
In the day's other tie, Everton got over their mid-week League Cup elimination against Manchester City by winning 3-0 at fourth-tier Carlisle United courtesy of goals from Arouna Kone, Aaron Lennon and Ross Barkley.
But the match was marred by racist abuse directed at Everton's players after Kone's second-minute opener, which prompted referee Lee Mason to ask for a warning message to be read out over the public address system.
"I thought it was dealt with in a very good manner," Everton manager Roberto Martinez told BT Sport.
"We agreed to make an announcement. I thought the referee was very good in how he dealt with that. It was just a reaction, something that we don't want to see."
Everton will visit Bournemouth in the fifth round.
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