All of City's goals came in the opening half-hour at Ashton Gate as they made a dream start to the season, with Bobby Reid netting a brace either side of a Famara Diedhiou strike before Barnsley scored a late consolation through Ryan Hedges.
Queens Park Rangers were the only other side to win by a two-goal margin as they overcame last season's playoff finalists Reading courtesy of a Conor Washington brace - the second goal of which came from the penalty spot after Tiago Ilori had been sent off for the visitors at Loftus Road.
There was late drama elsewhere in London as Nelson Oliveira rescued a point for Norwich City with an 88th-minute equaliser against Fulham, who had earlier taken the lead through Russell Martin's own goal at Craven Cottage.
There was no more than one goal in any of the other fixtures as the Championship season got off to a low-key start, but it was an unsuccessful return to the second tier for relegated Middlesbrough as they fell to a 1-0 defeat at the hands of Wolverhampton Wanderers at Molineux, courtesy of a Leo Bonatini goal on his debut.
Ipswich Town, Preston North End and newly-promoted Sheffield United also registered home wins by a solitary goal to nil, while Cardiff City were the only side to pick up an away victory as they beat Burton Albion through Kenneth Zohore's late winner.
Championship results in full: