The way the term "relief pitcher" has changed meanings so many times across the history of baseball, you get to see how effective pitchers were during their eras of playing major league baseball.
Back in the early days of baseball, there were hardly any relief pitchers. The starting pitcher would go the whole 9 innings, regardless of how they were pitching, and would throw up to 300 innings a season, sometimes more. Then when bullpens were starting to gain popularity in the 50's and 60's we saw the "fireman" come to light. A fireman was a guy who would come out of the bullpen and throw multiple innings to " put out the fire " and close out a ball game. And now in present-day baseball, you have one innings closers, one, sometimes two set-up men, maybe a left-handed specialist or a long reliever, but bullpens are paramount to a team's success today.
One way someone could track how a relief pitcher fared in their career is a statistical measure called WAR (wins above replacement) which shows how many wins a certain player is worth compared to a replacement player at the same position. It is helpful when determining how much value a player has not in their overall stats, but in every field that comes with a player, defense, hitting, pitching, and things like defensive positions and ballpark adjustments. The complicating thing with WAR though is it's ever-changing, the way WAR is calculated changes quite frequently so there are different outlets on how to calculate it as accurately as possible.
The Cardinals have had some legendary relief pitchers over their history, and there were some under-the-radar relievers that you'll see on this list that might surprise you. But let's first look at some pitchers that fell just short of making the top 5 best Cardinal relievers based on WAR.