Aybar, who has sat out the last five games with the injury, can come off of the DL on April 18.
"When you have an oblique, and that is an important part of what you do, you want to make sure he is far enough along and not set back," said Angels manager Mike Scioscia.
While Scioscia said the Angels may have been able to absorb the injury and given it a few more days, Angels doctors said Aybar would not be comfortable to play before next weekend.
Aybar will continue to run the bases this week and take ground balls at shortstop. He will also get in a game next weekend in the Minors before returning.
"He'll get in an extra day of work and he'll put it to good use," Scioscia said. "We are going to just give him the time."
In Sunday's game, Brandon Wood replaced Aybar at shortstop.
Rodriguez, a righty reliever, will be sure to help a depleted bullpen that absorbed a 14-inning game on Saturday night.
Using all seven relievers in the bullpen for the second time in four games, the Angels used a total of nine pitchers, including Dan Haren who got the win.
The nine pitchers also tied a club record for an extra-inning game.