The deal, running between 2012 and 2018, means Sky will broadcast all races, qualifying and practice sessions live on pay-TV, but the BBC will only have rights to half the races and qualifying sessions for free-to-air.
The BBC will broadcast all the "key races", including the Monaco and British Grands Prix, while the corporation will also carry highlights of any races that it does not show in full.
Formula One boss Bernie Ecclestone has hailed the "super" deal, saying: "There will be highlights as well as live coverage on two different networks now, so we get the best of both worlds."
However, the McLaren team principal Martin Whitmarsh has asked for clarification from Ecclestone on the deal due to fears that the sport could lose fans with only half the races on free-to-air TV.
Speaking to Autosport magazine, he said: "As I understand, the BBC are covering half the Grands Prix, and Sky are doing every practice session and everything else. It's interesting. I don't think anyone should be immediately reacting to say this is good, bad, or indifferent.
"What we need to understand is whether the large audience we currently enjoy in Formula
One will be maintained. I think we also need to understand exactly how this is being done."
Williams team chairman Adam Parr also gave a lukewarm welcome to the deal, but noted that the sport could benefit from the increased investment from Sky.
"In principle I have no issue with optimising the balance between the revenues that we need, and getting a good reach in the audience. The devil is in the detail," he said.
"I think it is a balance and, without knowing the details, you cannot comment on whether it is good or bad. What I do know is that Bernie is a very passionate believer in getting the broadest audience possible and I think he has almost certainly done this in order to do that."
Parr expressed sympathy for fans who would no longer be able to watch all the races live on free-to-air TV, but said that F1 was also an expensive business to run.
"I am sympathetic to them. I understand it is difficult - but English Premier League fans have had that for a while haven't they?" he told reporters at the Hungarian Grand Prix.
"The one thing I would say, which I have said before, I know that whether you are coming to a race or are watching the sport at home and have to do that on pay-TV which seems expensive, people have to bear in mind what it costs to put on this show. It is part of the character of F1.
"For us to design and build the two cars that we will have on the grid on Sunday here, without putting an engine in them, without putting a driver in them, without accounting for the 70 staff that we bring to each race - without all of that, those cars cost £2m. You multiply that by all the cars on the grid and that is £24m minimum of the costs just to make the parts. That is part of the show.
"It is not a bloke or two blokes with a tennis racket and a pair of plimsolls with zero cost. It is a very, very expensive sport. The best thing we can do for fans, whether they want to come to the races or want to watch it on TV, is to reduce the cost of the sport without spoiling the show."
Last month, it emerged that the BBC had decided against renewing its exclusive five-year deal to cover F1, which runs until the end of next season. It was understood that BBC bosses felt that the money spent on F1 would be better used elsewhere, as the corporation attempts to slash its budget by 20% under the new licence fee settlement.
In a statement today, BBC Sport director Barbara Slater said that the joint deal with Sky has "delivered significant savings" while also ensuring that live and extended highlights coverage of the sport "continues to be available to licence fee payers".