Mid is a little trickier. The advice I was giving was more towards the side lanes.
Winning mid involves blocking your lane properly, rune control, and waiting a bit. Trade damage with her often, but save up enough mana. Because what you'd want to do is get her just under your kill threshold, and vaporize her once that time comes. Hell, if you managed to pull off a great creep block from the get go, you can kill her for free.
For example, Ranter mentioned Storm Spirit earlier. For SS, this threshold should be level 3. At this point he can get two charges of Overload off with Vortex and Static Remnant. Combined with autoattack damage this is approximately 300 damage, or more than half of Drow's hp. So using what I said above, you'd want to trade blows until she's about 1/2 or 2/3rd hp. Then walk up, combo, and she's dead. Don't ever worry about your hp being too low. As long as you can survive 2 - 3 autoattack hits or about 150 hp you can kill her. Remember, she has no nukes. Her base damage is all there is.
In fact, if you want even more evidence that Drow is completely trouncable by you... just look at the stats of the melee creep.
The melee creep has more hp, movespeed, armour, everything! Than Drow. The only thing he loses is that the melee creep has half of Drow's base damage. If you do enough damage to kill a melee creep, you can kill Drow. Hell... since the melee creep has twice her armour and 100 more hp, you could kill Drow 30% more efficiently than a melee creep.
Seriously. Punch her in the face.