Lots of good Italian's in London, here's a few of my faves. Some are not that cheap, others very much so, none are dressy places and all serve authentic Italian. Not sure what you mean by the classics but none of these are a Luigis from the high street with packet pasta, packet sauces and garlic bread.
Bocca Di Lupo, my absolute favourite restaurant in London, brilliant food and a great experience. If there are just a pair of you ask for a seat at the bar in front of the chefs, the kitchen is in the middle of the restaurant and you can watch them at work.
http://boccadilupo.com/
The River Cafe, a London icon, twice daily menu changes, very seasonal, fresh and their pasta is amazing. Not cheap mind but the service is fantastic and very relaxing.
http://www.rivercafe.co.uk/
Zucca, like a cheap River Cafe, with a slightly more compact menu.
http://www.zuccalondon.com/
Trullo, again in a similar vein to the River Cafe
http://www.trullorestaurant.com
Polpo, a group of 3 very trendy Italian places, no booking, small plates and all that jazz
http://polpo.co.uk/