No. Nagato never lost to Rinnegan influence. Rather, he lost to hatred. See, hatred is what causes everyone to go out of control. Even Nagato said it during his conversation with Naruto. In uchiha cases, it is also hatred that is created by Pain that causes them to go berserk. You can see that afterwards, they can gain their cool and control the power. This is what Jiraya was teaching Nagato. All the advices and training were meant to help Nagato direct his power to do good./B]
So, in the manga there are two source of power - hatred and love. Naruto is one dude who gain power from loving others. Even so, due to great pain, Naruto was also overcame by hatred with the exception that he is bipolar - that is he is in control or not at all. In his case, Kyuubi was waiting for him to get hurt. It is because naruto can harness the huge power of Kyuubi in his right mind that Sasuke was in great respect for him.
So, they are not being overcome by the power of the dojutsus - rather, they got overcome by pain and hatred, and let their power go out of control.
