Coach Gary Kubiak said Sunday that the injury is "pretty severe," and that Tate will likely have surgery on Tuesday or Wednesday.
"It doesn't look good," Kubiak said. "That's the best I can describe it for now."
Houston plays its second preseason game in New Orleans on Saturday.
Tate's injury drops him out of a crowded competition at running back. Arian Foster started Saturday's game, and had four rushes for 31 yards.
Steve Slaton is Foster's main challenger after his 2009 season was cut short by a neck injury. He ran 10 times for 22 yards in Arizona, but fumbled on the goal line. Slaton had seven fumbles last season.
"It's tough, because that's what held us back and him back last year," Kubiak said. "To have it happen in the first preseason game, it's obviously disappointing. If it's going to happen, I'm sure glad it happened there, and not three weeks from now. But it's something that's got to get corrected for us to be successful."
Chris Henry, also in the mix for the running back job, had two carries for 6 yards and caught a pass for a 14-yard gain. Kubiak said Henry will move up the depth chart with Tate injured. Henry is also playing special teams.
"We're trying to get him a little more settled, but he's very aggressive," Kubiak said. "He showed up for (special teams coach) Joe (Marciano) and he will get more time at the back position, as we move forward with Tate being out. So we'll see, he did some good things."
Kubiak closely watched backup quarterback Dan Orlovsky, who completed 12 of 21 passes for 129 yards in two quarters.
The Texans signed Orlovsky as a free agent in March 2009. He was inactive for all 16 games last season, but has moved into the backup role behind Matt Schaub.
Kubiak liked what he saw from Orlovsky in Arizona.
"He was steady," Kubiak said. "Since he's been here with me, I thought it was his best outing in the preseason. He protected the ball, he moved his group, he had two big, long drives.
"He could still play better, and I think he will," Kubiak said. "It's important that Dan continues to show this team that he's taking steps forward and he took one last night."
The Texans advanced inside the Cardinals 20-yard line four times in the game, and came away with only three field goals. Their lone touchdown came on Schaub's 44-yard pass to Andre Johnson in the first quarter.
"We've got to do something better down there, we were 0 for 4," Kubiak said. "We just didn't finish down there. We didn't finish plays. We've talked about trying to run the ball better. It's a point of emphasis, and we'll go back to work on that."
The Texans' starting offense and defense looked impressive in the first quarter, building a 10-0 lead.
Schaub completed 5 of 6 passes for 78 yards and Johnson caught three passes.
Defensive end Mario Williams, slowed by a hip injury early in training camp, had two sacks, linebacker Brian Cushing forced a fumble and safety Bernard Pollard delivered a crunching hit on Arizona running back Tim Hightower before Kubiak called on his backups.
The Cardinals rallied from a 16-0 deficit to win in the fourth quarter, but Kubiak was encouraged by how well his first unit played.
"We've got to keep it in perspective," Kubiak said. "Our first group felt really good about themselves when they walked off the field. That's important, and that's all they got to play. That's not their fault they didn't play four quarters, that's on me."
Kubiak said receiver Andre Davis (bruised tailbone) and fullback Jack Corcoran (shoulder) had the only other significant injuries in the game.