Both clubs have agreed a transfer fee of £55,000 following a set number of appearances and a major percentage of any sell-on fee, should Fulham move the youngster on.
As a result of the agreement, The Robins have dropped their appeal over a ruling by The Football League, which stated that the Ashton Gate club were not entitled to compensation for the former club trainee.
The 19-year-old has been training with the West London club since the start of the season after rejecting a new deal with the West Country outfit and following the transfer agreement, Rosenior agreed personal terms with The Cottagers.
Robins chairman Steve Lansdown was pleased the affair had come to an end.
"This is not a satisfactory outcome, but it is a commercial decision bearing in mind the recent decision by a Football League tribunal that we were not entitled to any compensation," Lansdown told the club's official website.
"We are pleased that it has been recognised that Bristol City Football Club should be recompensed for our investment in Liam, and have got something out of the deal.
"It also shows that we were justified in standing our ground over the League's decision.
"As a result of the settlement we have passed on Liam's registration to Fulham. He is now their player and, hopefully, will go on to have a successful career."
Meanwhile, Bristol City have announced that they will play Manchester United in an Under-21 benefit match next month, with proceeds going towards the West Country club's academy set-up.
The match was arranged as part of a compensation package for 14-year-old midfielder Daniel Rose, who left The Robins to join United's famed youth academy in the summer of 2002.
