Goals from Pierre-Michel Lasogga and Kemar Roofe during the final quarter rescued a point for the West Riding club. By pegging back the sixth-placed Robins, Leeds ensured the gap between themselves and the promotion places remained at eight points.
Heckingbottom said: “We have got teams to play and I know it is a tough set of fixtures.
“But, while we are picking up points from the teams above us, no-one will be getting away from us.
“That is how we have to look at it. If we can pick up positive results we will see where we are before the last 10 games.
“Eleven points would have been big so, yes, we are still in there.”
United came close to snatching a victory with Lasogga heading against the crossbar deep into stoppage time.
That miss meant Heckingbottom became the seventh new head coach in a row to fail to mark his Elland Road bow with a win, Dave Hockaday remaining the last appointment to claim all three points on home soil courtesy of a 1-0 win over Middlesbrough in 2014.
The new Leeds chief added: “The third goal was going to be vital and we stressed that at half-time. Once we got it, I felt (a draw) was the minimum we were going to manage.
“We had chances right at the death. I know when I watch them back I will be cringing because I think they were really good ones. We possibly should have won it.
“Bristol City are really well organised at set-plays and it was those set-plays they played for. Because we didn’t deal with the threats and went behind, we played into an away team’s hands. That was the difficulty.”
Samuel Saiz will return for Leeds in their next fixture, Wednesday’s trip to Derby County, after serving a six-game ban for spitting. His return will boost United further after the manner of yesterday’s fightback.
“I am new into the club but I probably understand the club a lot more than some of the players who haven’t been here too long,” added the head coach. “Playing here, you’ve got to enjoy it and embrace it.”
Report: Page 2