Sir Alex Ferguson's side took the lead after just four minutes when Carrick poked the ball into the bottom corner after a scramble from a corner.
© AP Photos
Minutes later Patrice Evra found himself in space, but his effort went wide of the post.
Robert Huth had Stoke's first real effort of the game on 20 minutes, winning a header in the area, but he could only head wide.
Carrick had another attempt on goal from a corner, but his flicked header went over the bar.
Wayne Rooney came close with a free kick on 43 minutes, putting his curled effort just wide of the post from outside the area.
Stoke pushed forward late on in the second half, but United's defence stood firm to ensure they led at half time.