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 air 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 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 half the races only available on pay-TV.
In a blog posting last Friday, the BBC's head of F1 Ben Gallop accepted that there has been "considerable reaction" to the deal, but said that given the BBC's current "financial circumstances" the new agreement "offers the best outcome for licence fee payers".
In a string of angry comments in response to Gallop's post, fans vented their fury at the loss of Formula One on free-to-air television.
Formula One fan UKDelenn said: "Not good enough Ben. Not everyone can afford Sky, not with rising fuel and food prices.F1 has priced me out of going to Silverstone, now the BBC has priced me out of watching on telly. Not sure I can be bothered watching half a season. 2011 has been brilliant coverage from the BBC. After that, I am done (sic)."
Akcsl said: "This is a complete sell out. To watch the entire season in HD, as we do now, we, the licence fee payers now how to pay Sky twice, once for the sports channel and then again for HD. We are losing out to the greed of Sky and Formula One. Shame on the BBC and Bernie Ecclestone for this shoddy deal."
Rob Haswell commented: "Why can't somebody just admit that you've spent far too much on the Olympics and something has to give. Ask yourself this - do you think six million people are going to watch the Olympic torch being carried across Britain for 70 days? I think you could have found the money from somewhere else."
Matt_sefton said: "It's a really sad day for F1 fans in the UK. Personally, I watch all races and qualifying live. I can't stand highlights shows. In case nobody's noticed, there's a recession on, I've just taken a 10% pay cut and I can't afford (nor do I want) a Sky Sports subscription."
TheTickUK added: "F1 will lose millions of viewers and any credibility the current BBC production/presenting team have forged of the last few years. I have loved this sport for over 20 years and I increasingly feel that 2011 will be the last season I bother. (I hope the £900 million move to Salford and whatever the Olympics is costing is worth it.)"
A protest petition has been launched on the Petition Buzz website calling for Formula One coverage of all races to remain available on free-to-air television. As of this morning, the petition has attracted 20,705 signatures towards the aim of compiling 1m to heap pressure on the BBC.
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 last Friday, 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 F1 "continues to be available to licence fee payers".
The deal is expected to save the BBC around £30m a year, and the corporation claims that it would have had to drop Formula One entirely unless it had agreed to share the rights.