The meaning is clear.  The origin is not.  None of my usual sources offer much help.
First use of “on a roll” and any facts or good speculation about the origin of the phrase would be appreciated.

Thanks,
Cuchuflete

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

I too always assumed it referred to the (lucky) roll of a dice.

BTW I’m puzzled by the 1976 cite (or rather absence of one). I can’t find anything relevant under winning streak

That is odd—apparently they added winning streak to the list of phrases but forgot to add a cite.

Winning streak is further cross-referenced to streak, n1, 6b:

1976 New Yorker 15 Nov. 162/1 ‘We’re on a winning streak. We’re on a hot roll,’ one city official said happily last summer, during the triumphal series of events that began with Operation Sail and extended through other local Bicentennial celebrations.

Well--for example--in the game of craps, a sequence of good rolls could easily evoke such an expression.

Thanks to all of you for the good replies.