The Steelers released cornerback Isaiah Green to make room for Brown on the roster.
On what Brown's role will be with the Steelers, coach Mike Tomlin said, "Obviously his ability to take in and learn information is going to be an element of that. Obviously [we're] excited about acquiring a veteran offensive lineman."
A trade for Brown signals the end of the Mike Adams experiment at left tackle. Adams, a second-round pick in 2012, moved from right to left tackle in the offseason, and he has struggled to protect quarterback Ben Roethlisberger's blind side.
Jared Allen dominated Adams in the Steelers' 34-27 loss to Minnesota on Sunday in London, recording 2.5 of the Vikings' five sacks.
Tomlin acknowledged that the Steelers need improvement at left tackle, where Adams has started the first four games of the season.
"We've gotten below-the-line work and that's the reality of it," Tomlin said. "It needs to be above the line and we'll turn over any stone in order for that to happen."
Marcus Gilbert said he anticipates staying at right tackle, which is another indication that the Steelers are bringing Brown in to play left tackle.
"I think the coaches feel comfortable where I'm at now, Gilbert said. "I think sticking to the right side is the plan for me right now."
There are questions about how much of an upgrade Brown, who is signed until 2016 and has a $4.75 million base salary this season, would be over Adams.
The 6-foot-6, 324-pounder never lived up to expectations in Arizona. The Cardinals bypassed running back Adrian Peterson in the 2007 draft to bolster their offensive line.
Brown, a former Penn State star, started every game for Arizona in 2008-11 but missed all of last season because of a triceps injury.
Bradley Sowell is expected to replace Brown at left tackle for the Cardinals.
ESPN.com Cardinals reporter Josh Weinfuss contributed to this report.