France's relationship with their fans was strained after a dismal World Cup campaign in South Africa saw them suffer a first-round exit.
Coach Laurent Blanc replaced Raymond Domenech in a bid to lift Les Blues out of their crisis.
They slumped to a defeat by Belarus in their opening game of the Euro qualifiers but recovered after beating Bosnia and Romania by the same 2-0 scoreline.
"It was tense because for as long as it stayed goalless there was always the chance of losing to a free-kick or an individual mistake," Arsenal's Clichy told UEFA.com after France scored two late goals through Loic Remy and Yoann Gourcuff to defeat Romania.
"But the fans kept encouraging us and we turned in a strong performance. It was a beautiful evening for France.
"We messed up our first test against Belarus, but we've managed to get a win tonight. The fans just want to see us play good football and get three points, and we want them to get behind us.
"They were behind us from start to finish tonight and it was magnificent."
Euro 2000 champions France top Group D with six points from three games, one ahead of Albania and Belarus.