Tyler Clippard got the first crack at it, converting a perfect ninth inning in a 5-2 win over the Phillies.
The change came a day after Rodriguez nearly suffered his fourth blown save this year during a 2-1 win at Citizens Bank Park. It also didn't help that Rodriguez entered the day with seven earned runs in his previous eight innings.
Rodriguez will now be in a less stressful role, which could mean anything from pitching in the middle innings or working as a setup man.
"Henry has been great and not so great," Johnson said. "He might be trying too hard and I will probably change his role and put him in a less demanding role -- trying to get him back on track.
"As far as who my closer is going to be, it's going to be what I usually do in a ballgame. It will be who is fresh, where I like the matchups better."
With Rodriguez out, there is a good chance that Clippard, Craig Stammen and Sean Burnett will split time in the ninth inning.
Clippard, who has been the one of the best relievers over the last three-plus seasons, expressed to the team at the end of last year that he could close games. He even told his close friend Drew Storen that he would like to be given a chance to be a ninth-inning guy.
Johnson said that Clippard told pitching coach Steve McCatty on Tuesday that he wanted a chance to close. Clippard fanned a batter later in the day while earning his first save of the year and second of his career.
"There was a lot more adrenaline, a lot more nervousness, but it was a good feeling," he said. "I haven't had that in a while. So it was a lot of fun tonight."