In the age of Open Innovation, it is vital for a country in the lower middle-income bracket to free itself from those constraints which seriously weaken the links between science and industry. A descriptive analysis of these linkages in a post-Soviet economy—Armenia—sheds some light on developments in policy-making which reinforce the interests of the private sector in the academic research and development (R&D) sphere. However, the way of thinking is still predominantly the ‘science-push’ model—which is far removed from the (horizontal) Triple Helix concept. According to empirical analysis, the scarcity of innovative companies is a serious handicap for industry-science collaboration and if the private sector has little demand for knowledge or science, then the innovation system cannot be effective. Very few higher education institutions (HEIs) or research institutes have devoted attention to the management of technology transfer, including necessary human resources. Human capacity problems, outdated infrastructure and an ageing workforce are significant barriers in scientific organisations. The autonomy of scientific organisations is an important asset which only half-exists in Armenia. On the escape route from a command economy, there are two potential traps on the way of autonomy: one occurs when the state overarches legal autonomy and creates a semi-autonomous situation; the other arises when the state is reluctant to regulate the framework for autonomous scientific organisations. Both exist in Armenia.