Normally, only a small quanitity of blood is collected for tests, so my guess is that you weren't actually feeling the effects of blood loss (well, unless they actually did collect many tubes of blood). Many people also have a mental reaction to needles which manifests as physical symptoms. I normally don't mind needles, but one time, I was watching TV and a woman was having collagen injected into her lips. The camera did a close-up on the needle being stuck into this woman's lips over and over again and to my surprise, I soon ended up feeling weak and a bit nauseous. I was just watching the the needle on TV! Many people are similarly affected by the sight of blood or dead bodies and so forth. Even the brief pain of the needle being inserted can trigger certain physiologic reactions.

You are probably feeling dizzy due to low blood pressure. Whatever is actually causing the reaction, your blood vessels are relaxing, increasing the space available for blood in your ciruclatory system, and reducing the overall pressure. The classic method to deal with this lower pressure is to sit with your head down between your knees so your brain is at or below the level of the heart. Another method shown effective in clinical studies is to cross your ankles, then squeeze them, tightening the muscles in your legs and abdomen to increase the blood pressure to your brain. The flight suits of fighter pilots pulling high G maneuvers have air bladders that inflate and constrict their bodies to try and keep the blood from rushing to their legs and causing the pilots to pass out. Same principle.

I am not a doctor, so if you have concerns about this, please consult with a medical professional.