The Saints were beaten 2-0 by the Robins, allowing closest rivals West Ham and Cardiff to make up ground in the race to reach the English Premier League.
"I shall analyse our performance very carefully by studying the match DVD. I hate losing and so do the players so we won't be taking the result lightly," Adkins said.
"We may have lost to two deflections, but you have to give Bristol City a lot of credit. They are on a good run at the moment and the ground was really buzzing.
"It's important we bounce back quickly in our next game against Hull."
Ghana midfielder Albert Adomah opened the scoring for the hosts at Ashton Gate just after half time and striker Nicky Maynard applied the coup de grace to continue the Robins' impressive revival under new boss Derek McInness.
However, the win was not enough to lift City out of the relegation zone.
"There is a confidence about our play now and everyone is contributing," McInness said. "But we are aware that a slap in the face could be just around the corner."
West Ham cut the Saints' lead down to two points with a 3-1 win at home to Derby.
The Rams took the lead in front of 27,864 spectators at Upton Park, the biggest Championship crowd of the day, through Tamas Priskin.
However, goals from Carlton Cole, just before the interval, and Kevin Nolan and a Mark Noble penalty in the second half gave the Hammers victory.
"The performance was only good enough for 45 minutes," Hammers boss Sam Allardyce said.
"That needs to be better at home, especially, and it's something we want for 90 minutes from the players, starting Tuesday at Middlesbrough. It was a great comeback from 1-0 down, and in the end we ran out comfortable winners."
Cardiff cut their deficit on Southampton to five points, beating struggling visitors Nottingham Forest 1-0, with the in-form Joe Mason coming off the bench to head in in the 70th minute.
"The lads are starting to organise themselves on the pitch now and there are more voices out there than there was before," Bluebirds boss Malky Mackay said.
"It's about momentum and winning brings confidence."
Middlesbrough, two points behind Cardiff in fourth, were held to a 1-1 draw at Peterborough, where Paul Taylor curled in an impressive goal with nine minutes left.
Visitors Barnsley grabbed their first victory at Elland Road since 1990 as they beat hosts Leeds 2-1 in a feisty Yorkshire derby.
Birmingham snatched a 2-2 draw at Blackpool, with Serbia striker Nikola Zigic salvaging a point with three minutes left at a breezy Bloomfield Road.
"I nearly got blown away driving to the game and for two teams to produce as good a game as that in those conditions was fantastic," said Blackpool boss Ian Holloway.
The point was enough to edge Blackpool into the last play-off spot ahead of Hull, who conceded three times in the last 12 minutes to suffer a 3-2 defeat at home to Burnley.
Leicester held hosts Portsmouth to a 1-1 draw to spoil Michael Appleton's first match in charge of Pompey.
Reading grabbed a dramatic 3-2 victory at Ipswich with two goals in the 90th minute to consign the Tractor Boys to their fifth straight defeat.
Crystal Palace and Millwall played out a goalless south London derby at Selhurst Park.
Brighton beat sorry Coventry 2-1 while rock-bottom Doncaster held lowly visitors Watford to a goalless draw.
Source: AFP
Source: AFP