In order to close a tab in the classic console, you have to use Salesforce Console Integration Toolkit. The toolkit contains closeTab() that allows you to close the tab.
The challenge here is you can only use Salesforce Console Integration Toolkit from VF page and not lightning component. So you have to find a way for your VF page to close tab.
To communicate between lightning component inside a VF page we use Window.postMessage.
So when save is completed, you have to post a message from lightning component to VF page. The VF page will have a listener to it, and you will then call
closeTab() method for SF toolkit to close the current tab.
I havent personally tried it, but in theory it should work.