The hosts held a slender 13-11 lead at the break, with Bristol wing David Lemi responding to Will Hurrell's try.
But second-half scores from Ross McMillan and Ben Glynn put the away side, who finished the season top of the table, in command of the tie.
The two teams play the second leg at Ashton Gate on Wednesday, 25 May.
Bristol, who are taking part in the Championship play-offs for the fifth time in seven seasons, elected to play the first leg away from home after losing in the final in each of the last two campaigns.
They are looking to return to the Premiership for the first time since they were relegated in 2009, when they finished bottom with only two wins from 22 matches.
Andy Robinson's team finished 16 points ahead of second-placed Doncaster in the regular season.
Doncaster: Jarvis, Bulumakau, M. Clark, Hurrell, Lewis, Cusack, Heaney, List, Hunter, Quigley, Challinor, Phelan, Makaafi, Hills, Shaw.
Replacements: Veikoso, Brugnara, W. John, Steadman, Young, Flockhart, Field.
Bristol: Arscott, Varndell, Tovey, Mosses, Lemi, Sheedy, Cliff, O'Connell, McMillan, Perenise, Evans, Glynn, Mama, Lam, J. Phillips.
Replacements: Brooker, Traynor, G. Cortes, Sorenson, Robinson, Roberts, Morgan.

Natalie Putt left her home in Thornleigh, Lower Gornal, on 1 September 2003, leaving her 11-week-old son at the house with relatives.
It was the last time anyone reported seeing the 17-year-old.
West Midlands Police said the case remained open and any credible information would be investigated.
Det Sgt Matt Morgan said: "It remains a missing persons' inquiry as there is no evidence she's been murdered and certainly nothing to suggest she was suicidal at the time she vanished."
Natalie had been out with a friend the day before she went missing.
An 18-year-old man was arrested in 2004 in connection with her disappearance but was later released without charge.
Police sealed off private land in Wombourne, south Staffordshire, earlier this year after someone recalled seeing a man digging near stables where Natalie is understood to have kept horses.
Det Sgt Morgan said: "Forensics teams carried out extensive searches and archaeologists used specialist radar equipment to reveal any buried items, or [reveal] whether land had been disturbed.
"Nothing suspicious was uncovered though and the search was called off after three days."
Natalie was described as white, of slim build, and just over 5ft (1.5m) tall  She was wearing a pendant necklace with the words "someone special" engraved on it.
Bristol took a significant step towards promotion to the Premiership with victory at Doncaster in their Championship play-off final first leg.
[NEXT_CONCEPT]
Information is being sought over a teenage mother who went missing in the West Midlands 10 years ago this month.