Enter Time Machine from backup of another Mac

I need to access multiple Safari passwords that were saved in my keychain, but which I recently deleted. I have an external hard drive with a backup of my previous MacBook Air (sold). Is it possible to enter the backup in Time Machine on my new computer, or otherwise just extract the passwords I need? I don't have space on my hard drive to backup my new MacBook Pro, restore the MacBook Air backup onto it to access the passwords, then restore the MacBook Pro backup...so I would have to buy another external hard drive to accomplish this.