The 21-year-old has made more than 100 first-team appearances for United, scoring 11 goals, including six this season.
The Scotland Under-21 skipper was called into the full national squad for the recent friendly victory in Norway and has also captained his club.
Armstrong told the club's official website: "I am enjoying my football, scoring goals and improving as a player so signing the extension was the obvious choice.
"It is such a joy being involved in our games and all the players have a similar vision to the manager as to where we can go as a unit, so I look forward to the next few years being very exciting for United and our fans."
The news is yet another reason for United fans to be cheerful following an explosive run of form by their team, who have scored 17 goals in four matches.
Manager Jackie McNamara said Armstrong had produced some "sensational" performances this season and added consistency to his game.
"There are not many players who can play the game the way he plays it and he excites the fans," McNamara added.
"He is an exceptional talent who will grace a larger stage at one point but his decision to commit to a longer contract at Dundee United shows that he knows this is the best place to continue his development."
Highly-coveted teenagers Ryan Gauld and John Souttar recently extended their contracts along with forward Nadir Ciftci.
McNamara added: "There is a real positive vibe around the club and this will show the United fans that the players are committed to taking this club where it wants to be."