If you want to do it with a scope all by itself, you will need a scope that can do trace math, and then average the results of the math operation.
I've attached a scope capture showing the grid voltage and the current drawn by a compact fluorescent lamp. The CFL has a switching supply inside and draws very spiky current waveforms.
The scope is set up to show grid voltage on one channel and lamp current on another channel. The math function multiplies the two waveforms, and the the result is averaged.
The yellow trace is the grid voltage (120 VAC here in the U.S.), the green trace is the lamp current (measured across a 1 ohm resistor in series with the lamp), and the purple waveform is the product of the voltage and current.
The average of the purple waveform is shown at the bottom of the image; it's 23.8 watts.
If you can extract the captured waveforms from your scope, you could do the multiply and average with Mathcad, or some similar program.
To safely make a measurement like this, you need to use an isolation transformer, or use isolated differential probes, which is what I did.
Another method to measure the true power is to use a wattmeter.
The second image shows the reading on an electromechanical wattmeter, showing 25.9 watts. The meter is only a 5% accuracy meter, so the reading is the same as the scope within the accuracy of both instruments.
Another possibility is to use a Kill A Watt meter:
Amazon.com: P3 International P4400 Kill A Watt Electricity Usage Monitor: Electronics
which, as you can see, can be had for about US$20. I have one, and it reads the power of the CFL as 26 watts.