Hi Gaurav,
According to your description, I would like to know if you need to use the Authentication.asmx web service in SharePoint Online.
However, the Authentication.asmx service is not available in SharePoint Online. Here is a similar question and replies in the following thread which you may find it helpful:
http://community.office365.com/en-us/forums/153/p/20137/94520.aspx
Since our forum doesn't focus on the development, we can only provide limited assistance with it.
However, you can post this question in the SharePoint Development Center forum and get more assistance with it.
SharePoint Developer Center forum: http://social.msdn.microsoft.com/Forums/en-us/category/sharepoint
In addition, I have found some useful links for your reference:
Remote Authentication in SharePoint Online Using Claims-Based Authentication
http://msdn.microsoft.com/en-us/library/hh147177.aspx
Developing SharePoint Online Solutions
http://msdn.microsoft.com/en-us/library/gg454741.aspx
Thanks,
Wilfred Ying