Your Arduino won't be able to handle more as 5 volt and 40mA per pin.
But you can control NPN-transistors or N-channel mosfets using your arduino, which can handle a much higher voltage/current. Which type of transistor may depend on the length/ nr of leds on your strip.
This instructable shows you how it's done.
Apparently the writer only used a 12 volt power-supply, which created the need for a 5volt regulator for the arduino as well, but you won't need it if you already have a 12 and 5 volt supply.