There have been claims that Dennis Eckersley was revolutionary, but in truth the closer was already a cliche by the time Tony LaRussa started using Dennis Eckersley almost exclusively in the ninth inning. LaRussa's use of Eckersley was only slightly more extreme than the Twins use of Ron Davis or how dozens of other closers had been used over the previous ten years. What made Eckersley unique was the level of success he had. The result was that many managers started to focus on using their closers for only one inning and found that it increased their level of success. With this record of success, most teams today have a pitcher who is used almost exclusively one inning at a time to finish games.
There are those who argue that this is not the optimal use of the best reliever. But they base this largely on the assumption that the success of modern day closers does not depend on how they are being used. Historically there have been some players like Mike Marshall that have had outstanding success while pitching in the 100 inning range in relief. But these appear to have been the exceptions. The prevalence of successful closers today depends on their limited use for short stretches.