The odds are you aren't going to get that 2:1 revised up to a 1st unless you find some evidence that you were wrongly marked down and should have had higher marks. They won't change the boundary just for you.
However given that your undergrad was at Imperial and your masters is at Imperial, you have a chance of being accepted anyway on the strength of your 2:1, rather than the 1st. The reason is because it's the same institution, the Masters admissions tutor is going to have trust in the quality of your course and can easily contact your undergraduate tutor for an honest view on your ability.
Most Masters course ask for a 2:1, now if the one you have applied for is asking for a 1st this means one of two things, either the course is very demanding and lots of people fail it, so they have upped the entry requirements to make sure only the best get in, or the course is oversubscribed and they have upped the entry requirements as a filter otherwise the class sizes would get too big.
If it's that the course is very demanding but they have places available, then they may be persuadable to give you the benefit of the doubt here given that you were so close. As long as you are self-funding (or have funding from an external source) then it is you who bears the risk not them so it's in their interests to give you a shot at it.
If the issue is that they are oversubscribed then they are likely to be harsh and say if you don't get a 1st we can't take you.
But you can take a chance on it being the first option and see what they say. I used to work in a university postgraduate admissions department where the standard offer was a 2:1 and often you got people with a 2:2 asking to be considered, we generally had room for a few more on the course so the admissions tutor would scrutinise their module marks, often ask to speak to them to weigh up their attitude and more often than not they gave them the benefit of the doubt - it was extra income for the University of course. However we often found that the next question the person asked was "unfortunately I don't have funding, is there any chance you could give me a full fee scholarship?" at which point the answer rapidly became sorry no.