Sometimes when I have negative feelings, it seems that a deep breath resets these feelings. Does that really happen? Is there any research on the effect of taking a deep breath on reducing the experience of negative emotions?
Yup; basically, it works. Varvogli and Darviri (2011) review research on diaphragmatic breathing, reporting:
Deep breathing has been successfully used to decrease the fatigue associated with haemopoietic stem cell transplantation patients 55, to reduce the anxiety and asthma signs/symptoms of children with asthma 56, in the management of acute stressful tasks 57 showing that the slow-breathing technique can have a significant effect on improvement of the hemodynamic changes following the acute stressful tasks. Furthermore, it has been used to influence autonomic functions in patients with essential hypertension and thus reduce it 58–59, in the management of male adolescent aggressive behaviour 60, in long term prophylaxis of migraine 61, in stress related to dental visits 62.
Fatigue, anxiety, stress, and aggression are all negatively valent (see the diagram in this answer; aggression doesn't appear, but is a form of negative arousal). It wouldn't be surprising if effects on hypertension and migraine incidence are partially mediated by emotional changes either.
Varvogli, L., & Darviri, C. (2011). Stress management techniques: Evidence-based procedures that reduce stress and promote health. Health Science Journal, 5(2), 74–89. Retrieved from http://www.hsj.gr/volume5/issue2/521.pdf.
As cited within Varvogli & Darviri (2011):
$55$. Sang-Dol, K., & Hee-Seung, K. (2005). Effects of a relaxation breathing exercise on fatigue in haemopoietic stem cell transplantation patients. Journal of Clinical Nursing, 14(1), 51–55.
$56$. Chiang, L. C., Ma, W. F., Huang, J. L., Tseng, L. F., & Hsueh, K. C. (2009) Effect of relaxation-breathing training on anxiety and asthma signs/symptoms of children with moderate-to-severe asthma: A randomized controlled trial. International Journal of Nursing Studies, 46(8), 1061–1070. Retrieved from ResearchGate.
$57$. Nogawa, M., Yamakoshi, T., Ikarashi, A., Tanaka, S., & Yamakoshi, K. (2007). Assessment of slow-breathing relaxation technique in acute stressful tasks using a multipurpose non-invasive beat-by-beat cardiovascular monitoring system. Conf Proc IEEE Eng Med Biol Soc, 5323–5325.
$58$. Mourya, M., Mahajan, A. S., Singh, N. P., & Jain, A. K. (2009). Effect of slow- and fast-breathing exercises on autonomic functions in patients with essential hypertension. Journal of Alternative and Complementary Medicine, 15(7), 711–717.
$59$. Kaushik, R. M., Kaushik, R., Mahajan, S. K., & Rajesh, V. (2006). Effects of mental relaxation and slow breathing in essential hypertension. Complementary Therapies in Medicine, 14(2), 120–126.
$60$. Gaines, T., & Barry, L. M. (2008). The effect of a self-monitored relaxation breathing exercise on male adolescent aggressive behavior. Adolescence, 43(170), 291–302.
$61$. Kaushik, R., Kaushik, R. M., Mahajan, S, K., & Rajesh, V. (2005). Biofeedback assisted diaphragmatic breathing and systematic relaxation versus propranolol in long term prophylaxis of migraine. Complementary Therapies in Medicine, 13(3), 165–174.
$62$. Biggs, Q. M., Kelly, K. S., & Toney J. D. (2003). The effects of deep diaphragmatic breathing and focused attention on dental anxiety in a private practice setting. Journal of Dental Hygiene, 77(2), 105–113.
Nick's answer is very thorough. I would add that the physiological symptoms of high-stress (muscle tightening, accelerated heartbeat, shallow quick breath) can be reversed by controlling breath voluntarily. So, when you change these physical conditions, emotional state often follows.