Driver opened the scoring in the 11th minute, sending a free kick from 25 yards out right of the middle inside the far post.
"It was a big game for the club," Driver said. "Obviously, with a team like Stoke, it is nice to score in a game like that. It's nice to get a win and a goal, but I would rather get that on the weekend (against Chicago in MLS play)."
Barnes scored in the 88th minute, taking a pass from Corey Ashe in the left box and sending a shot along the end line from outside the left goalkeeper's box and inside the near post.
"We created some really good chances in the first half," Houston coach Dominic Kinnear said. "I thought we defended well. Then it was a bit of a possession game for them, and we were kind of dropping deeper and deeper and had some good possession as well. Giles had a good individual play for the second goal."
Stoke were playing its first game under new manager Mark Hughes. Stoke will play FC Dallas on Saturday night and end their three-game tour against the Philadelphia Union on Tuesday night.
"From our point of view, this is exactly what we need at this stage of the season and our preparation," Hughes said. "I thought for long periods we showed decent control of the football without really having a cutting edge. We didn't create enough in the final third. We didn't really retain the ball as well as we hope to do in the future, but for a first game, first test, I thought this was exactly what we needed."
Stoke defender Geoff Cameron, who spent 4½ years in Houston before he moved on to the EPL with Stoke, said the first game under Hughes was a good opportunity to get back on the field.
"It's obviously a preseason game, so we are just rolling with it," said Cameron, a U.S. national team member. "It's a new system, so we are just trying to get the ball down and play. Obviously, everyone was a bit fatigued playing in the heat, but I thought it was a good show."
Cam Weaver had a line drive from the right box in the 23rd minute stopped with a leg save by Stoke goalkeeper Asmir Begovic. Jason Johnson let one go from inside the upper box in the 33rd minute, but Begovic made a diving save to keep it 1-0.
Barnes nearly upped the lead to 2-0 in the 69th minute, but his right-footed one-timer from 10 yards out on the right box just missed the far post.