Full-back Watson, 22, made his first-team debut in January 2007 and has now made 78 appearances, overcoming several serious injuries along the way to establish himself as a regular this season.
Scotland Under-21 midfielder Armstrong, 20, has made 47 appearances for United's first team.
Manager Peter Houston said: "Both players fully merit these extensions. Keith has been part of the first-team squad for a few years now and this season has progressed even further. He knows he has still to learn and develop parts of his game and it is great that he wants to continue that development here at Tannadice.
"Stuart has shown already this season that he can play a large part in our team and make an impact in games.
"He is an exciting player that brings additional pace and skill into our team. He still has some parts of his game to refine and I am delighted that he sees Dundee United as the best place to do so."