I understand it can feel weird to negotiate when the salary offer is substantially above what you've been making, but I second that it's well worth doing, and that a job offer is very unlikely to be retracted on the basis of negotiating.
My first experience with this came with a position I didn't intend to negotiate. The manager told me on hiring that they 'really couldn't do more than [salary higher than anything I'd ever made in my life]', and that no one at the same rank in the organisation made more than what they were offering me. What did I know? I happily took the salary offered. Fast-forward six months later, and the same manager was doing a handover so I could manage his position while he was on annual leave. As an incidental part of the handover, we were going over a spreadsheet that included everyone's salaries - and, who'd a thunk it, others at the same rank were being paid more... :-P. I didn't even have to say anything - just looked at him - and he stuttered that he'd get right on an amendment to my contract...
My next position - which I needed badly at the time I took it, and which pays well even at low levels, but which unfortunately I loathe... I was already holding the position on a contract basis when they offered it to me on a permanent basis - along with a $1000/month reduction in salary. A good portion of this was probably 'fair' given the difference in benefits and security for ongoing work - it was also in an area where ongoing positions are incredibly rare. But frankly, I hated the job so much that I just wasn't going to do it for less takehome pay... So I held out for the higher salary, and also insisted they remove the huge probationary period (two years!) that was attached to the original offer, but that they still pay for the extra credentialing that they normally only provide to probationary staff...
Now, I really was seriously ready to walk (while it was hugely financially advantageous, there's still a significant part of me that wishes they'd said no...), so I'm not sure what I would have done if I'd actually wanted the role. But the whole experience has made me hugely more aggressive in trying my luck - which, in the place I'm working now, has actually had a flow-on effect to other female staff, many of whom we're putting up with some pretty ridiculous contract clauses and default pay arrangements until I negotiated mine. There's now a much more level gender playing field in this particular office as a result...