City were on course for a club-record 10th consecutive victory in all competitions that would have seen them move to within a point of Premier League leaders Chelsea, who drew with Southampton earlier in the day.
However, after George Boyd pulled one back, Barnes completed a memorable comeback for Sean Dyche's side, who were 2-0 down at half-time thanks to strikes from David Silva and Fernandinho at the Etihad Stadium.
Burnley had offered signs of promise in the opening exchanges, but City's dominance in possession came to fruition when Silva opened the scoring 23 minutes in.
The Spaniard's fifth goal in as many Premier League games was soon followed by an impressive second from Fernandinho 10 minutes later as the home side appeared out of sight.
However, Burnley had a lifeline early in the second half as Boyd converted from close range, despite appearing offside, and the fightback was complete when Barnes fired home from inside the area nine minutes from time.
It could prove a vital point for strugglers Burnley, while City boss Manuel Pellegrini - again without skipper Vincent Kompany following a hamstring injury - was left to rue a missed opportunity to keep the pressure on Chelsea.
While Jason Shackell was required to block Martin Demichelis' goalbound effort after five minutes, Burnley acquitted themselves well early on.
However, City continued to dominate possession and were almost ahead when James Milner scooped Jesus Navas' delivery over the crossbar.
Unchanged Burnley enjoyed their best spell of the half as Kieran Trippier's delivery evaded Barnes before Dean Marney stung the palms of Joe Hart from the edge of the area after 19 minutes.
A flash of City's quality enabled them to grab the lead, however, as Navas' cutback found Silva, who turned smartly and fired past Tom Heaton at the near post.
The playmaker's recent form has helped City cope with the continued absence of Edin Dzeko and Sergio Aguero, with Fernandinho - starting ahead of the rested Yaya Toure - getting in on the act shortly after the half-hour mark.
The Brazilian midfielder was afforded too much time and space by the Burnley defence and responded by curling a stunning strike in off the crossbar from the edge of the area to open his account for the season.
Dyche's mood will have been lifted a minute after the interval when Boyd diverted Danny Ings' cross home to give the visitors hope. Ings almost got on the end of Barnes' throughball shortly after, but failed to get it under control.
And, just when it seemed as though City would see the game out, Barnes drew Burnley level.
Barnes half-volleyed his fourth Premier League goal of the season when City failed to clear a free-kick to gain his side a reward for their spirited second-half showing, halting the hosts' momentum in the process.
For City, it represents the first time they have failed to beat a newly-promoted side at home since November 2009, when they were held 3-3 by Burnley.
