Eva-01 is always under Yui's control theory.

Throughout the show, we see multiple times how Evangelion moves by her own will, even when she's not even activated. She saved Shinji from the falling ceiling in the first episode, and she broke from the Bakelite in the EoE JUST after Asuka was eaten. And it is for a reason, I believe. Yui knew that Shinji needed to go crazy so everything worked, so she waited for Asuka to die first (It's not my idea, I saw it on YouTube and a lot of people, including myself, think it's true.).
So, how does she do that? Like, want really really much? I don't think so. I think Yui was aware, that if she will do too much stuff by herself she would have troubles (Like it was in episode 22 when Gendo didn't want to let Shinji pilot Eva-01 because it was dangerous and used her after only because there were no other way.)
She understood that, and she tried not to do anything if it isn't necessary. I guess berserk takes more time for her to activate because Eva literally has no energy, but in a normal state, she can move immediately.
What do you guys think? For me, it seems pretty reasonable. Also, if you would ask "why only she has that power", I think that could be because Eva-01 was the only Eva with a full soul, not split ones like in Eva-02 or Eva-00 (Theoretically, part of Lilith's soul, because it's the most popular theory that I also believe is true).