Twice two goals behind, the Tigers roared back to lead 5-4 late on and were only denied victory by Joe Bryan's equaliser deep into six minutes of stoppage time.
Adkins said: "We wanted to be in control of our own destiny today and it is great to achieve safety on the back of a positive result.
"Again the players have shown great character and a real sense of togetherness to twice come back from two goals down. Some of our attacking play was outstanding.
"Of course, I am not happy to concede five and there were some really poor ones from our point of view. But overall we have come out of a brilliant game to watch, having not been beaten by a team desperate to get into the play-offs.
"Now we want to continue building on our recent form in the last two games."
Harry Wilson fired Hull ahead from just inside the box, but by half-time they trailed to Marlon Pack's 37th-minute shot from distance and a cool 40th-minute finish by Famara Diedhiou.
A Diedhiou header on 53 minutes put the hosts 3-1 up only for goalkeeper Frank Fielding to concede a 56th-minute own goal when a header from Hull substitute Fikayo Tomori hit a post and went in off him.
Bobby Reid then slotted his 21st goal of the season to make it 4-2 but Wilson beat Fielding at his near post with a 30-yard free-kick from the left and substitute Abel Hernandez fired home from close range to level.
When Fraizer Campbell converted a low cross from substitute Kamil Grosicki on 84 minutes it seemed Hull would pull off an amazing win. But Bryan had the last word deep into added time, shooting home from the edge of the box.
Bristol City head coach Lee Johnson said: "We are all a bit shell-shocked. Having played well in the first half, we got ourselves 4-2 up and in a great position to take the points we needed.
"Our attacking play deserved even more goals but some of the defending was schoolboy errors.
"We will keep fighting, but obviously we now need a storming finish to the season and hope things go our way.
"Perhaps we lack enough leaders on the pitch. Players can be divided into followers, standard-setters and leaders and we need more to graduate from the second of those categories to the third.
"That was my message in the dressing room after the game. It was a fantastic game for the neutral, but not the outcome we wanted."