There are several workarounds for this issue.

1. The most simple in this case is to use YYYY-MM-DD.

So we would have to change line

component.set("v.record.Date__c", '2016-1-1' );

into

component.set("v.record.Date__c", '2016-01-01' );

2. Another possible workaround is mentioned here:

We could add line

rec.Date__c = new Date( rec.Date__c );

just after

var rec = component.get( 'v.record' );

line in perform method of javascript controller.

3. Another general workaround for a custom date population was described here.

There are several workarounds for this issue.

1. The most simple in this case is to use YYYY-MM-DD.

So we would have to change line

component.set("v.record.Date__c", '2016-1-1' );

into

component.set("v.record.Date__c", '2016-01-01' );

2. Another possible workaround is mentioned here:

We could add line

rec.Date__c = new Date( rec.Date__c );

just after

var rec = component.get( 'v.record' );

line in perform method of javascript controller.

3. Another general workaround for a custom date population was described here.