Swansea City were denied just a second win at Ashton Gate in 27 years as a late penalty from Nahki Wells cancelled out Jamal Lowe’s opener.
In a fixture that has historically been low scoring, Lowe netted early in the second half with a sweeping left-footed strike.
It put the Swans on track for a third win in four away games this season, only for Wells to score with eight minutes to go after Connor Roberts had been adjudged to have fouled Antoine Semenyo.
The result extended the Swans’ unbeaten away run in league fixtures to 10 games, but Cooper and his squad will be frustrated having coped with any threat from the hosts with relative comfort.
The head coach had made just one change from the side that started the draw at Coventry, with Jay Fulton coming in for Viktor Gyokeres.
As had been the case with a number of recent meetings between these sides, the opening exchanges were tight with little in the way of goalmouth action.
Roberts produced a good delivery from a corner, with Daniel Bentley swatting it away for Bristol City, while Jake Bidwell headed wide from another set-piece at the midway point of the half.
The hosts responded, with Chris Martin getting away down the left before Marc Guehi made an important intervention to cut out the cross.
Bidwell just failed to make a connection with Korey Smith’s deep centre as the game began to show signs of opening up.
Jamie Paterson worked Freddie Woodman after picking off a loose Roberts pass, while Andre Ayew smashed a fierce shot at Bentley that the keeper parried into the air before having penalty appeals waved away as he went down trying to reach a Bidwell cross.
Woodman denied Tomas Kalas from an excellent Chris Brunt delivery just before the interval as the first half ended goalless.
But the stalemate was broken five minutes into the second half by Lowe.
Some neat build-up allowed Grimes and Roberts to combine and slide the forward in on goal, where he produced a powerful finish across keeper Bentley into the far corner.
Guehi made a superb tackle to prevent Martin having a tap-in from a Semenyo cross as the Robins sought an immediate response, while Ayew worked Bentley at the other end as the Swans broke with purpose.
Dean Holden’s side were dealt a double blow as they lost Andreas Weimann and Steven Sessegnon to injury in quick succession, with Callum O’Dowda and Famara Diedhiou their respective replacements.
The visitors came close to doubling their lead with 15 minutes to go. Ayew produced a mazy run before moving the ball on to Smith and Roberts.
The latter fizzed an excellent ball into the six-yard box, but keeper Bentley held on as Ayew – having continued his run – sniffed a close-range finish.
But parity was restored in debatable fashion when referee Oliver Langford adjudged Roberts had fouled Semenyo as he fell when defending a set-piece.
Substitute Wells made no mistake as he drove the spot-kick down the middle.
Viktor Gyokeres flashed a volley across the face of goal for the Swans, and hopes of finding a winner were raised when seven minutes of added time were confirmed.
They created a clear opening from Woodman's accurate kick downfield, but Smith snatched at the chance after being fed by Gyokeres and the ball flew wide of the target.
Bristol City: Daniel Bentley; Zak Vyner, Taylor Moore (Nahki Wells 78), Tomas Kalas (captain), Chris Brunt; Steven Sessegnon (Famara Diedhiou 73), Andreas Weimann (Callum O’Dowda 65), Jamie Paterson, Jay Dasilva; Antoine Semenyo, Chris Martin.
Substitutes: Max O'Leary, Jack Hunt, Tommy Rowe, Tyreeq Bakinson.
Swansea City: Freddie Woodman; Ben Cabango, Ryan Bennett, Marc Guehi; Connor Roberts, Korey Smith, Jay Fulton, Matt Grimes (captain), Jake Bidwell; Andre Ayew, Jamal Lowe (Viktor Gyokeres 76).
Substitutes: Steven Benda, Ryan Manning, Wayne Routledge, Liam Cullen, Yan Dhanda, Kyle Naughton.
Referee: Oliver Langford