United were the first to create a chance when Wayne Rooney let fly from 35 yards out, but his shot was saved by Asmir Begovic at the bottom left-hand corner of the goal.
Rooney had another great chance a few minutes later when Patrice Evra produced a teasing cross, which allowed the England striker to shoot on goal, although the ball crept just too wide to the left of the post.
Robin van Persie was next to try his luck just before the half-hour mark when he got onto the end of a headed pass from Phil Jones, but his effort was blocked on its way to goal.
In a half of few chances, Stoke tried their luck just before the break as Erik Pieters let fly with an effort that flew high and wide of David de Gea's goal.
United found themselves a goal behind in the 38th minute when Adam lashed in a delivery towards goal, which perhaps took a slight deflection off of Michael Carrick to elude De Gea and give the hosts the lead.
United were dealt a further blow just before the break when Jones was stretchered off after a heavy fall, with Carrick having to shuffle into defence as a replacement centre-back.