The simple answer is: you MUST use an analog filter in front of your ADC. The reason (in one word): aliasing.
By its very nature, a digital filter cannot distinguish between a signal frequency less than the Nyquist limit (half the sampling rate) and one which is greater.
Let's take an ADC with a 1 kHz sample rate. Look at a DC signal. The output of the ADC will be a DC level, right? Now look at a 1 kHz sine wave (perfectly matched in frequency to the ADC). You'll get one sample per cycle, at the same point in the waveform, every time. This will be completely indistinguishable from a DC level.
Now let's look at a 999 Hz sine. Each successive sample will look at the sine at a slightly different point, and the result will be indistinguishable from a 1 Hz sine wave. Same for a 1001 Hz signal. If you don't filter out everything above 500 Hz before the ADC samples, you will be dealing with signals which are not the frequency you think they are.
Once you've got a clean set of sampled data, it's true that digital filters can give better performance than analog, and particularly if you need to vary the filter parameters on the fly digital can do much better than analog.
But first you've got to get clean data. And in principle, the input must be filtered to reduce the level of unwanted signal to less than the resolution of the ADC (assuming you want filtered data to be accurate to that level). This requires a lowpass filter with very, very steep cutoff, and very good low-level performance. In your case, if you're using a 12-bit ADC, and you're looking at signals which have full-scale components above 5 MHz, you need a filter which will have a gain of 1 at 1 MHz and 1/4096 at 5 MHz. That's a very steep cutoff, and the filters used are generally termed "brickwall filters". Of course, if your input signal components are less than full scale above 1 MHz, the filter doesn't have to be as good, so don't get too freaked out. And to make life easier, oscilloscopes generally don't need more than 8 bits of resolution, so that helps too.