Absolutely!
Although it will be much tougher without a First, because you will be competing against those people that did (although obviously grades are not the whole story). Coming from a humanities background I can say that funded PhD positions are extremely competitive, lots of people apply, and lots of people inevitably lose out. This had been my experience, because although I did have good marks (First undergrad and 2:1 masters), there were many other people out there with better grades, better connections to relevant academics, and/or (initially at least) better practical/industry experience. Following my Masters I was trying to get on funded programmes (with the PhD questions already developed by academics) for about 5 years. I would apply to one or two per year whilst working full time, dedicating a ton of time researching and writing proposals. I got interviews to about half of all those I applied for, but was always unsuccessful. In the end I was successful and managed to get on an industry funded PhD. I think I was successful here because I developed my own research idea instead of trying to fit into an existing programme, and because I developed a good set of contacts in industry and academia.
This approach worked for me because: 1) my grades were slightly weaker than they would be for other candidates, 2) it had been several years since my Masters so I would not have the opportunity of being 'shepherded' or guided into a programme, and 3) because my practical/industry background and my relationships with industry professionals had grown to be much stronger than my connections to universities and academics.
I don't know how it works with engineering, but I think its completely possible for you to do it. You just need to keep the faith and be resilient because it may take a while. Get a good research idea, and build up relationships and connections. Good luck!