Martin Luther King, Jr. said that "the arc of history is long, but it bends toward justice." John Maynard Keynes (later plagiarized by Winston Churchill) said that "[people] will do the rational thing, but only after exploring all other alternatives." I believe that they both are correct: Reason and social justice prevail in the long run, though we seem to take the longest rather than shortest route possible. So the facile answer to the subject-line question is: Be patient.
Of course, much suffering occurs in the meantime, and there is never any guarantee that humanity will survive long enough for the arc of history, and the lathe of trial-and-error, to work their magic. So the question becomes: How do we accelerate the "natural" processes that bend the arc of history toward reason and social justice?
It helps first to understand those processes.
There are two related concepts which are most useful for understanding why Martin Luther King, Jr. and John Maynard Keynes were correct in their respective observations about the tendency toward the eventual triumph of reason and justice over ignorance and injustice. British biologist Richard Dawkins coined the term "memes;" cognitions which, like genes, are packets of information which self-replicate (through communication), mutate (through interpretation, synthesis, and innovation), compete for reproductive success (in individual choices of what to believe and what techniques to utilize, which aggregate into social institutions and technological regimes), and thus evolve.
American Philosopher of Science Thomas Kuhn, at about the same time (the mid-1960s), coopted the term "paradigms" to refer to sets of memes which create a systematic lens through which to view reality. In his view, once a dominant paradigm in any given science emerges from the chaos of competing views, it focuses investigation within that paradigm, allowing the subsequent accumulation of anomalies (findings that are incompatible with the paradigm) to usher in an eventual paradigm shift through attention to and resolution of those anomalies. Though Kuhn was particularly referring to the formal process of scientific investigation, the same process occurs in a more haphazard way in informal cultural evolution.
We have several mathematical tools for studying these interrelated processes: microeconomic analysis, including game theory; network analysis, including epidemiology (the study of the spread of contagions, whether in the form of viruses or memes) and "cellular automata" (a way to model complex interactions among nodes of a network); and complex dynamical systems analysis (modeling feedback loops and threshold effects).
Complementary to these highly abstract tools, we have at our disposal a very useful typology of social institutional modalities: The conceptual breakdown of the undifferentiated social field into hierarchies, markets, norms, and ideologies (see my essay "A Framework for Political Analysis" for descriptions of, and comparisons among, these modalities). These are simultaneously products and manifestations of the underlying social dynamical abstractions quickly surveyed in the preceding paragraph.
What I have just outlined is a paradigm, in the Kuhnian sense, one which creates a context for focused investigation, certain to yield telling anomalies and give way to a fruitful paradigm shift. Most of the useful insights generated by the current chaos of competing theoretical frameworks (e.g., Marxist analysis, pure microeconomic analysis, functionalism, and even post-modernism) are subsumed by this paradigm, if you examine it closely enough, while most of their obfuscating noise is left on the cutting room floor. It is, of course, an imperfect paradigm, but you have to start somewhere, and this is the best we have to work with at present.
So what does all of this academic esoterica have to do with defeating conservatives? The answer is simple: To the extent that thoughtful people are successful in focusing our intellectual efforts through this (or some similar) paradigm, we are simultaneously successful at shortening King's "arc of history" and accelerating Keyne's "[exploration] of all other alternatives." As I've written elsewhere, the ultimate political battlefield is the human mind, and those who best utilize their own minds best advance our collective cause on that shared field of combat.
The paradigm I've sketched out (and that is fleshed out in thousands of volumes of academic publications) not only helps us to understand what is, but also helps us to understand how to affect it to more effectively create what might be.
Sometimes we forget what an important role thought plays, even long before it is ever translated into action. We all admire, for instance, the impressive successes of Mahandas Gandhi and Martin Luther King, Jr., both of whom implemented effective strategies of passive resistance. But we often forget how much both of them owed to Henry David Thoreau, who implemented very little in his life, but articulated the ideas that Gandhi and King drew upon.
The politics of mud wrestling over blind ideological certainties, and shouting back and forth our respective zealous but relatively superficial convictions, accompanied by obsessions with personalities (both of participants in the exchange, and of public figures) and horse races (overly reducing this epic battle on the field of human consciousness to relatively unimaginative political rituals of strategic interaction), can only take us so far. The field eventually belongs to those who go beneath the surface of political combat, and learn the currents that flow beneath it.