The home side were undone by the sort of set-piece goals that infuriate managers, with the opener coming after 34 minutes when Mee climbed above Luke Ayling to powerfully head home a left-wing corner from David Jones.
And when Jones delivered another flag kick from the right six minutes later, it was again a Burnley player who showed most determination in the air as Keane nodded past Ben Hamer from close range.
City kept battling to the end and netted a consolation when Jonathan Kodjia was given credit for forcing the ball in from close range in injury time as Burnley failed to clear a right-wing cross.
Steve Cotterill’s hosts should have been ahead before the opening goal, having played some delightful attacking football prompted by young midfielder Bobby Reid.
He had a shot brilliantly blocked by Keane on 15 minutes and then set up striker Kodjia to have a 21st-minute effort kept out by Mee’s intercepting header.
Aden Flint sent two good headed chances from Reid crosses straight at goalkeeper Tom Heaton, who also saved well from two more Kodjia shots.
Burnley were largely content to catch the home side on the break, with striker Andre Gray, who moved to Turf Moor after turning down City, being roundly booed by Robins fans every time he touched the ball.
And it was the visitors who took control before the interval, with first Mee finding the target and Keane then extending the lead after 40 minutes to silence the home support.
Sean Dyche’s team looked stronger defensively in the second half, with Keane and Mee continuing to impress and the midfield taking more of a grip on their opponents.
City were running out of ideas, although centre-back Derrick Williams drilled a 70th-minute shot against a post from 25 yards with a sweet left-footed strike.
Aaron Wilbraham also had a shot deflected into Heaton’s arms while, at the other end, substitute Lukas Jutkiewicz fired wide from distance.
Flint had another header saved by Heaton from an 88th-minute corner before Kodija finally pulled a goal back, but City were unable to repeat the late heroics that earned a point from their previous home game against Leeds.
Last season’s League One champions have taken only one point from their opening three Championship home games and have shipped poor goals in all of them.
Burnley won without having to produce much in the way of penetrating attacking play, but looked well-organised and sent their 1,200 travelling home fans back to the north west in positive mood.