Your p-value of .01 implies that the variable is significant at 99%. That's "good." The low R^2 doesn't have any bearing on the validity of that result per se, it just means that there's a lot of the variance that isn't explained by your included variables.
Now, this could be because there's just a lot of random error. But it could also be because there are other variables that matter, but that you haven't included. To the extent these variables may be correlated with your variable of interest, your results could be subject to omitted variable bias.
Your burden is the following: Either you have to defend theoretically the proposition that there are no other explanatory variables and it's all random error, or you have to show that any such omitted variables aren't correlated with the variable of interest. A third option would be to include them in the regression and redo the estimation.
Showing or proving a lack of correlation may not be as challenging as you might think--if for example your variable of interest is a "treatment" and you have a randomly assigned control group, any missing variables would be uncorrelated with the choice of group and hence with the treatment.
Whether your results are "significant' in the non-statistical sense is up to you to decide--but a low R^2, again, does not inherently challenge the conclusion that your variable of interest affects the dependent variable.