Sinatra fought like a tiger for the part of Maggio in the film---a gamble which (paid off) with an Oscar.
请问为什么不用was paid off ?

pay off : 这里是 “（取得）成功”的意思。
Did your plan pay off?
你的计划成功了吗？
Our plan certainly paid off; it was a great idea.
"我们的计划当然成功了,那是个很好的主意。"

Sinatra fought like a tiger for the part of Maggio in the film---a gamble which (paid off) with an Oscar.
Sinatra 在影片中扮演Maggio这个角色, 他同老虎搏斗 --- 一场冒险最终成功赢得了奥斯卡。

这里which引导定语从句， pay off 的真正主语是a gamble, 赌博冒险获得成功，得到奥斯卡。是主动语态。而不是冒险被奥斯可成功。