If we are speaking generally, yes. A dual power relationship, as Lenin saw it, can be practiced by anarchists and be considered an anarchist approach.
However, as Lawrence pointed out, dual power relationships are spontaneous developments rather than part of some sort of movement building strategy, as Lenin used the concept. What the term means to the anarchist vanguard is pretty much the same as turning Civil Society into a political instrument against the state. http://en.wikipedia.org/wiki/Civil_society
In this strategy, the "dual power" fulfills the same space as "general strike". It is a myth that people work towards achieving, even when its truth and effectiveness may not actually exist without embracing this myth.
There are implications that Civil Society doesn't need government, business or mafias...but this is the same as the libertarian pro-capitalist view that a world of business can exist without government. These views can easily become political and can easily be recuperated by approaches like democratic socialism, which desires pretty much the same thing, but is willing to work through the government to achieve it.
A follow-up question and perhaps the real answer being sought is: Can a dual power relationship end social alienation? No. Dual power relationships rely on institutions that reify struggle, so any society built through institutions will not achieve an anarchy. Dual power relationships are massive when observed. The democracy used in such relationships may be less like a republic, but they also do anything but free power. Power is condensed inside various delegate councils, which historically condense power in similar ways to a republic and create reasons to play politics, where we talk about what we'd like to do and hope for agreement rather than do what we want.
There is no great vision coming from those that seek a society that is dominated by civil institutions. Today's unions run what was business while charities, churches and non-profits take over the social services traditionally offered by the government as well as continue to offer those services that they always have, minus the dollar bill being a factor.