It sounds like you're bumping into the issue that the "Canon EOS Flash Bible" calls "EOS Flash Photography Confusion".
Whenever you take a flash photo, you have two main sources of illumination: the ambient (i.e., all the existing light in the scene that isn't flash), and the light coming from the flash. The two sources combine in any flash image you're taking, and can be balanced against each other to taste however you want. You can have an image be mostly flash with no ambient (brightly lit subject, black background) up to mostly ambient with a small bit of flash (aka "fill flash"--because the flash "fills in" the shadows).
Ambient is controlled by iso, aperture, and shutter speed (this you know). Flash, however, is controlled by iso, aperture, flash-to-subject distance, and the flash power setting.
The camera doesn't know how you want to balance the flash against the ambient, but makes a "best guess". So the behavior changes with the shooting mode.
In M mode, you're in full control. How you've set your iso and aperture and shutter speed control your flash/ambient balance if your flash is in eTTL. If your flash is in M mode as well, you can control the flash power output directly, too.
In Av and Tv mode, the assumption the auto-exposure system makes is that you want flash for fill. So, the camera settings will be made based on the ambient metering, and will be pretty close to what you would have had without the flash. So, in low-light situations (which is where most people begin using flash), Av can yield very slow shutter speeds (unsafe for handholding) and low levels of illumination.
In P and Auto modes, this behavior changes to something more like how P&S cameras behave. The flash/ambient balance will be set to fill in good light levels, but in lower light levels, the balance will shift to mostly flash so that the background can be black, but the subject will be well-lit, and handholding shutter speeds are always used.
My suggestion to you, assuming by "correct exposure" you mean P&S flash behavior in Av, is to look at your camera's custom settings, and change the shutter speed used for flash in Av mode. Shutter speed is your one independent control for flash/ambient balance, because it will affect your ambient, but not your flash. The faster the shutter speed (up to your maximum sync speed), the more illumination in the image comes from the flash. Most Canon bodies have the ability to force the shutter speed in Av to be your sync speed, or from 1/30s-x-sync (i.e., handholding safe speeds), which will then make it behave more like a P&S flash.
However, if you really want to learn flash and balancing ambient against the flash, putting your camera in M mode and your flash in M mode, and reading the following two articles on flash vs. ambient, may be the fastest way to learn.