Tries from Manu Tuilagi and Foden put England 14-3 ahead and when Tom Croft scored late on, after a French revival, they led 24-15 with just five minutes remaining.
Wesley Fofana’s converted try in the corner brought France storming back into the game but Francois Trinh-Duc’s late drop goal attempt fell short and England had won a thrilling victory.
Lancaster is currently in charge on an interim basis but this famous win in Paris can only enhance his application to lead England into the 2015 World Cup on home soil.
Asked whether he would like to see the coaching team remain in place, Robshaw said: “Yes. They have created a great environment for us to go out there and play and of course as players we will back them to keep on doing it and keep on improving.
“I would like to think they think the same of us. We have only been together for 35 to 40 days now and we feel we have come a long way since that game against Scotland.”
Foden, who has scored in all four of his Tests against France, added: “They have been marvellous so far. The thing that Stuart has done is that he has been honest with all the players and the media.
“He has done what he set out to do and created an environment and culture that is certainly positive.
“All the boys are behind him and we all buy into what he is trying to build here and everyone is enjoying their rugby and enjoying the camp.”
England, mathematically at least, will head into the final weekend with a shot at winning the Six Nations title.
Never before have England won all three away fixtures in the same championship season and yesterday’s triumph lifts Lancaster’s men up to fourth in the world.
“It is there for all to see what Stuart has done. The biggest achievement, without a shadow of a doubt, is he has put some pride back in the England jersey,” Farrell said.
“That is not just the squad but throughout the country. You can feel the wave of support behind you.
“Everyone is enjoying the ride with us. That gives us massive belief.
“To get a side together, a new side with a lot of inexperienced players, as quickly as he has done speaks volumes
“Stuart always said we are building team for the future, but it is also about now.
“We had some doubters along the way. We had two good wins and against a Welsh team on fire we acquitted ourselves well in that game.
“Then we were told we will see what the team is about in France and to come here and play like they did and fight like they did, you cannot understate how big an achievement that is.”
Lancaster would not be drawn on the recruitment process and his chances of landing the permanent job – but England’s performance in victory can only strengthen his case.
France had not lost at home in the Six Nations in 10 matches and Lancaster said: “I am very proud to come to the Stade de France and get a win, where few sides have done that recently.
“It’s not about me, Andy or Graham. It is about the team. We lacked experience in terms of caps but we have a huge amount of self-belief in the direction we are going
“The players deserve great credit. I am hugely proud of the performance they all put in. It epitomised the character we have in the side.”
“Great credit to the French as well. When Tom scored they never gave up and put us back in a position where they could have dropped a goal to win the game.
“I am pleased with the boys’ resilience.”
France scorers: Tries: Fofana. Cons: Parra. Pens: Beauxis 3, Dupuy, Parra.
FRANCE: C Poitrenaud; V Clerc, A Rougerie, W Fofana, J Malzieu; L Beauxis, J Dupuy; J-B Poux, D Szarzewski, N Mas; P Pape, Y Maestri; T Dusautoir (capt), J Bonnaire, I Harinordoquy.
Replacements: F Trinh-Duc for Beauxis (73), M Parra for Dupuy (49), V Debaty for Poux (49), W Servat for Szarzewski (49), L Picamoles for Bonnaire (66).
England scorers: Tries: Tuilagi, Foden, Croft. Cons: Farrell 3. Pens: Farrell.
ENGLAND: B Foden; C Ashton, M Tuilagi, B Barritt, C Sharples; O Farrell, L Dickson; A Corbisiero, D Hartley, D Cole; M Botha, G Parling; T Croft, C Robshaw (capt), B Morgan.
Replacements: B Youngs for Dickson (73), M Stevens for Cole (68), T Palmer for Botha (55), P Dowson for Morgan (62), R Webber for Dowson (74).
Referee: Alain Rolland (Ireland).