The best draft steal by the Cardinals franchise (considering both Arizona and St. Louis) has to be a pair of St. Louis Cardinals players that were drafted in the seventh round or later.
Larry Wilson was a safety for the St. Louis Cardinals that was drafted in 1960. Wilson was elected to the NFL Hall of Fame in 1978. Wilson was a great draft day steal for the Cardinals because he wasn't drafted until the No. 7 round of the 1960 draft, where he was the drafted No. 74 overall. Wilson played for the Cardinals from 1960-1972.
Before you think that being drafted No. 74 isn't such a steal, which would be a round-three draft choice now, consider that the NFL had only 12 teams that drafted in 1960, which is why he went in round-seven.
Wilson was a dual threat at safety, in that he either was blitzing the quarterback from his safety spot, or he was picking off a pass, as Wilson wound up with 52 total interceptions in his career. Wilson was drafted as an offensive player out of college, but the Cardinals thought he could play at cornerback. That didn't work, so they tried him at safety in a preseason game, and when he was asked to blitz, Wilson showed that he excelled at the move, and the rest as they say is history.
Wilson was named to eight Pro Bowl games and to All-Pro seven times in his career. In 1966, Wilson intercepted passes in seven straight games. He wound up leading the NFL with 10 interceptions that year. In total, he returned five of his interceptions for touchdowns over his career. The NFL didn't keep track of sacks back then, so we don't know how many he was responsible for in total.
According to his entry in the NFL Hall of Fame, Jackie Smith was a tight end for the St. Louis Cardinals that was drafted in the No. 10 round of the NFL draft of 1963, and was drafted 129 overall that year. That was quite a draft day steal fro the Cardinals, considering that Smith went on to have a Hall of Fame career.
Smith was a 6'4" tight end that weighed 235 pounds. He played for the Cardinals for 15 years, and then hung around for one more year, when he played for the Dallas Cowboys in 1978. Unfortunately, he is mostly remembered for his one year with the Cowboys, due to a dropped pass in the end zone in Super Bowl XIII, despite all the years of excellent play with the Cardinals.
Smith was named to All-Pro teams twice and five Pro Bowl teams. When he retired following the 1978 season, he was the all-time leader in receptions for a tight end with 480 catches for 7,918 yards and 40 touchdowns. Smith had a career average of 16.5 yards per reception.
His best game came in his rookie year against the Pittsburgh Steelers, when he gained 212 yards on nine catches. Smith also doubled as the Cardinals punter for his first three years with the team. Smith was elected to the Hall of Fame in 1994.