Jevons rose above defender Karl Broadhurst in the 70th minute to score with a header from a centre from the left by Lee Johnson.
Bournemouth and Bristol City served up plenty of entertainment but there were precious few chances for the home team's biggest crowd of the season.
Bournemouth had plenty of early chances with Adriano Basso getting down in the second minute to keep out a low header from James Hayter.
A minute later and a centre from the right by Stephen Purches ran along the top of the crossbar before dropping to safety.
Bournemouth included former England winger Darren Anderton for his second appearance since signing on a pay-as-you-play basis.
Anderton found the hectic pace too much at times but he showed his quality after 36 minutes with a fierce free-kick from 22 yards which Basso got down to save.
Anderton lasted 76 minutes - a five minute improvement on his previous appearance - before coming off.
By then Bournemouth were trailing to Jevons' fifth of the season and there might have been another.
Seven minutes after half-time Scott Brown unleashed a fierce right-foot shot from 25 yards which hit the inside of a post before coming out.
Bournemouth substitute Leon Best came close to equalising in the 87th minute when he went around Basso but from a difficult angle shot into the side-netting.
City had three players booked as Bournemouth went in desperate search of an equaliser.
Louis Carey was the first to earn a yellow card for a foul on Hayter but Jamie McAllister was lucky to stay on for a piece of retaliation in the 82nd minute when he lunged at Broadhurst.
Jevons' night turned sour also in the 89th minute when he kicked the ball away after a free-kick had been awarded against him and saw yellow.
But City held on to the delight of the 1,000 travelling fans who witnessed their team take another giant step towards promotion.