Antonio Conte's side were swatted aside by their London rivals in September, going down 3-0 at the Emirates on the weekend after Liverpool's win at Stamford Bridge.
The two consecutive league reverses left some questioning Chelsea's title credentials, but Matic believes they have acted as a catalyst rather than a hammer blow.
“You know sometimes you need to lose the game to wake up,” Matic said. “The Arsenal game was a wake-up call for us. We knew that we had to do better, to want more, and we did that.
“It’s easy to say we needed to work harder but after Arsenal we knew we had to show that we had quality, and we have done that in the last three games. We have shown quality, we don’t concede, and we score goals.
“I was thinking about [defeats to Arsenal and Liverpool] before the game,” he added. “This was a moment to show we can compete with every team. I’m very happy because of that.
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“Of course we don’t forget that we lost against Arsenal, against Liverpool. But of course the table shows that we are there, we are near. We will try and continue like this.”
The victory over United has taken Chelsea to 19 points, one off leaders Manchester City, who are level with Arsenal and Liverpool on points.
Having initially threatened to make the season something of a procession, Pep Guardiola’s side have not won in three. Arsenal, meanwhile, dropped points at home to Middlesbrough this weekend, while Liverpool’s defensive frailties persist.
Matic knows that a prolonged run of good form from him and his team-mates could topple the early pacesetters.
“Every game is very hard,” he said. “City started very well until the last few games when they dropped points, which is good for us. It’s a long season. So we will see what is going to happen until May. Every game is a different story.”
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