Since January 2016 you can use your Contactless or Oyster card to travel from Gatwick to London on both normal rail services, and the Gatwick Express.
The prices for single fares on normal services can be found in the Single Fare Finder, and based on it a trip from Gatwick to East Croydon costs £5.20 peak / £3.00 off peak, to London Bridge (or other southern rail terminal like Victoria) it would cost £14.20/£8.10 and across the city (Rail + Underground including travelling in Zone 1) £16.50/£10.30.
Note that based on these prices it's apparently cheaper to go from Gatwick only until East Croydon for £5.20 peak, exit the station, and re-enter to continue until Victoria or London Bridge for £5.10, as that would only cost £10.30 in total (compared to £14.20 for the trip without splitting the journey in Croydon)
Caps also apply, the daily cap is £31.00 peak and £19.30 off peak for any London zone. Apparently there is also a weekly contactless only cap of £102.20, which is actually a few pounds cheaper than a weekly rail season ticket would cost between Gatwick and London zones 1-6.
Prices on the Gatwick Express however are different, they cost £19.80 at any time, and don't count towards any of the daily and weekly caps.
Note that Gatwick was already added to the TFL rail map
Since these prices are not all available to download in PDF form, and you have to use the single fare finder, I made a small table that shows some of the possible combinations of stations to/from Gatwick Airport.
Destination Peak price Off-peak price
Horley £1.70 £1.30
Salfords £2.60 £1.80
Earlswood £3.50 £1.90
Redhill £3.70 £2.00
Mertsham £4.20 £2.30
Zone 5,6 (including East Croydon) £5.20 £3.00
Zone 4 £8.30 £5.90
Zone 3 - Overground stations £9.30 £5.90
Zone 3 - Rail stations £9.40 £7.60
Zone 2 - Brockley or New Cross Gate £9.80 £6.60
Zone 2 - any other station £11.60 £8.10
Zone 1 - Rail only £14.20 £8.10
(including trips across London Z1-6 on Thameslink)
Zone 1 - Rail + TFL £16.50 £10.30
(including trips across London Z1-6)
Zone 1 - Rail + TFL when £14.00 £10.30
switching at Elephant and Castle
Zone 7 - Carpenders Park £18.20 £11.80
Zone 8 - Watford High Street £18.90 £12.00
Note that the prices are not completely based on zones inside London, as some trips are cheaper than others, even in the same zone.
Prices up to date as of 23/05/2017 - Prices were unchanged between Gatwick and Croydon since 2016. There were £0.10-£0.50 increases on the rest of the network. They also fixed some of the weird structures where travelling further on the Overground into Zone 1 was actually cheaper than getting off at Zone 2.