Speaking from experience here (i've passed out from injections before), it isn't easy to overcome fears like this but you can do it. The fear will come when you're thinking about it too much, so try to push those thoughts away when they come, by doing things like:
1. Focus on breathing, in through the nose out the mouth, slow and controlled breaths, envision breathing out your fears while you exhale.
2. Divert your attention, simply counting to 10 repetetively can do the trick, and don't look directly at the needle. Something i was taught by a counsellor was to try and imagine yourself in a bubble of safety, there are lots of tricks/methods out there.
3. Sometimes if the person is nice they will talk to you to take your attention away, maintain that conversation if you can. Perhaps even ask if they can keep talking to you beforehand, to take the anxiety away.
4. Give yourself some kind of reward for going through it, be it choccy or something like that. For me it would be a bottle of e-liquid for my vaping
It is something you have to work at, it may not work initially, but with practice stuff like this does help; mind over matter.