The Armenia international scored his fourth goal of the season in United's 4-0 fourth-round victory over Wigan Athletic, meaning the Red Devils are now the only English club left competing on four different fronts.
Jose Mourinho's men are 12 points behind Premier League leaders Chelsea, but recently set up a League Cup final meeting with Southampton and remain in the Europa League with a last-32 meeting against Saint-Etienne to come in February.
"We are taking all the competitions seriously," Mkhitaryan told MUFC TV. "For Manchester United, every cup is very important. A trophy is a trophy.
"It doesn’t matter if it is the League Cup, the FA Cup or the Europa League. We want to do our best to try and win them all."
Like Jurgen Klopp, Mourinho too fielded a number of younger players in his FA Cup fourth-round clash, handing competitive debuts to Axel Tuanzebe and goalkeeper Joel Pereira.
And Mkhitaryan believes that both players "did well" and will benefit from their experience at Old Trafford.
"It’s great for them to play their first game," said the former Borussia Dortmund star.
"You know as a player that, when young players come into the squad, you have to help them and offer them some advice and I thought they both did well on what is a great experience for them."