univariate significant variables only for multivariate binary logistic?

I have to do a binary logistic regression with one categorical dependent variable and several categorical and continuous independant variables. When i enter all these independant variables together i get several significant variables. However, if i run a one way anova initially for each independent variable separately and then select only the significant ones for the binary logistic then very few are significant. My doubt is which method is corrrect?. Several articles are selecting only the significant variables on univariate for the multivariate. There is literature at the same time to suggest that all the variables should be entered. Could you clear this dilemma as regards which method i should use?