Why do humans need emotions? If the emotions' only purpose is to stimulate certain behavior, why can't this task just as effectively, or even more effectively be done by the reasoning part of the brain?

Are emotions really necessary for survival?
This question becomes more complicated if we think in terms of "emotions" (e.g., angry, happy, sad, afraid, etc.) than in terms of "affect" (positive and negative feelings, high and low arousal).  I'll start with affect and move on to emotions.

An affective state tags an object with a certain value--and it does so very quickly (e.g., Pham, 2007). For example, if a snake makes you feel bad, then it must be undesirable. This motivates a plan of action that could involve elimination of the snake (fight) or withdrawal from it (flight). You do not have to spend time considering the value of the snake (i.e., that it's bad and dangerous) and developing a plan of action to deal with it. Instead, your negative affect facilitates automatic evaluation and action, which might increase your evolutionary fitness. Gerald Clore has suggested that negative affect serves a STOP function whereas positive affect serves a GO function.

Emotions are a bit more complicated, in part because we don't have a consensual definition of them.  If emotions are merely socially shared concepts (happy, sad, angry) that we use to describe and organize our situated affective experiences (e.g., Barrett, 2014), then we can say that emotion concepts serve self-regulatory and social-communicative functions.  For example, by labeling as anger my high arousal and negative affect in the context of having my autonomy violated, I've organized this multifaceted affective experience into a single concept (of anger). This enhances my capacity to develop ways to regulate my affective experience (Kashdan, Barrett, &amp; McKnight, 2015) and it facilitates communication of it.  If I tell you that I am angry, then you'll have a general idea of what I'm feeling.  Self-regulation and social communication are certainly important for survival.

Emotions are involved in lots of other things, e.g., facilitating cooperation (e.g., DeSteno et al., 2010), motivating the pursuit of new goals (Bench &amp; Lench, 2013), social bonding, deceptive behavior, cheating, and so on.  All of these can be adaptive in certain contexts.

It's also important to note that emotions are flexible. For example, we don't have a "fear module" in the brain that motivates a rigid pattern of action (e.g., Lindquist et al., 2012).  Instead, emotions (on one view) emerge from the interactions of more domain-general processes (including exteroceptive perception, language, interoception, conceptualization, executive attention, and core affect; Oosterwijk, Touroutoglou, &amp; Lindquist, 2015).  These domain-general processes interact to produce emotional episodes with patterns of cognition, behavior, physiology, and experience that are flexible to the situation at hand (Barrett, 2015). The kind of flexibility and efficiency that you might expect from the "reasoning part of the brain" probably does not match up to the kind of flexibility that I just described.  Also, it's important to note that emotion and cognition are not separable constructs.  

EDIT:  As far as reason vs. emotion is concerned, emotion is a much more potent motivator.  If you consider seizure patients with lesions to the majority of their amygdalae (from lobectomies), they don't show fear.  If you put them in a dangerous situation, they won't run away--despite having their prefrontal areas intact. (Although an alternative non-emotional interpretation is that these patients have lost the ability to discern what is salient, e.g., threatening, in their environment.)  If you have a hobby that you really enjoy, your motivation to continue doing it is largely emotional ("It makes me happy").  As onigame implied in his/her answer, emotions motivate these kinds of decisions.

Anyway, I think I've provided you a very incomplete view on the adaptiveness of emotions, but hopefully this clarifies some things.