Thanks Olivier for giving some helpful tips.
Just for information to any one who might need this:
I have solved this issue by learning how to assign the tickets to an Agent and thus setting the tickets to open by default. Saying that I mean once we assign the new tickets generated from the Social Media Channel these tickets are by default as Open for the Agent to whom these tickets are assigned.
Assigning the tickets can be done by setting the Work Distribution for either Agent or for Organizational Level (to assign it to a level of team as per requirement). More information on same can be found in the help center provided on Cloud for Customer tenant. Also one can refer to Escalation of the tickets to higher level of team.
Cheers,
Chandan.