There are many carpet cleaners that you spray, agitate, then vaccum up when dry. Personnally I think extraction is far better...and you don't need a fancy carpet extraction machine...merely a wet/dry vaccum.
Vaccum well, spray a good cleaner, work with a brush, spray with warm water and vaccum with a wet/dry vaccum. Rinse (spray water) and vaccum again.
Having written that, some time ago now we got new carpet in the house and sometime after that got a Bissel Proheat 2x. I used it in the car the first time for my recent spring detail. I used it on the cloth seats as well as the carpet and carpet mats. You would not believe the dirt (dirty water) that came out of what appeared clean and when it all dried out, it looked brand new - seriously. I'm not saying you should run out and buy one of these extraction contraptions, but if like me you have one (or access to one) anyway...well doh.