Nothing as to origin in OED but I’d be astonished if it proved to refer to anything other than the roll of a pair of dice.
roll, n.2 1,h. In colloq. phr. on a roll (orig. N. Amer.), enjoying a sequence of successes or a run of good fortune.
1976 [see winning streak s.v. WINNING vbl. n.1 9]. 1979 Tucson Mag. Jan. 26/1 Now she is 26 and on a roll. 1983 Christian Science Monitor 5 Dec. 43/2 The paranoia of seeing the Soviets on a roll, the sense of the U.S. in decline, has about evaporated. 1984 Times 21 Mar. 19/2 The economy is on a powerful roll, but I am not worried about overheating. 1985 New Yorker 29 Apr. 55/2 Culpepper was on a roll… He could do no wrong.
BTW I’m puzzled by the 1976 cite (or rather absence of one). I can’t find anything relevant under winning streak