The visitors were fully rewarded for their attacking intent, but it took a late save from keeper Ben Smith to deny City an equaliser and the Rovers' stopper was grateful to see an even later Mark Bridge-Wilkinson free-kick come back off the woodwork.
It would have been cruel luck on Rovers if City had claimed a draw after Sean O'Driscoll's men had bossed most of the game and should have run out more convincing winners.
Heffernan, who celebrates his 26th birthday on Friday, gave himself an early present when he instinctively volleyed home a superb right-wing cross from James O'Connor after 14 minutes.
Rovers had already shown their desire and intent by that stage and with their midfield trio of Mark Wilson, James Coppinger and Paul Green on top it seemed they could run away with the contest.
Wilson, making his first start since joining Rovers last month, impressed with a composed display in front of his back four while lively duo Coppinger and Green worked overtime to dominate City's midfield.
Rovers' keeper Smith hardly had a touch in the opening period and the only worry for O'Driscoll's men was the fact that they couldn't find that vital second goal to kill off the contest.
Winger Jonathan Forte impressed with his pace, but he wasted several good openings as Rovers' tried to convert possession into end product.
Eight minutes before the break he left Mark McCammon fuming when he beat his team-mate to Gareth Roberts' teasing cross only to blaze his shot over the bar.
Injuries and suspension have disrupted Bradford's efforts to climb the table and manager Colin Todd had to shuffle his limited resources again.
With playmaker Bridge-Wilkinson playing out on the left they struggled to find any route through Rovers' well-drilled backline and it was not until the closing stages that they tested the visitors' resolve.
Heffernan will be disappointed he didn't double his tally when he tried to lob keeper Donovan Ricketts from the edge of the box and there were headed chances for skipper Graeme Lee and Adam Lockwood before Bradford's late rally.
On-loan full-back Nathan Doyle was stretchered off with a leg injury which could scupper his hopes of an extended stay at Valley Parade, but his mood would have been improved had City's late push salvaged a point.
In the final minute of normal time Dean Windass eventually got some space in the Rovers' box, but his low drive was brilliantly pushed round the post by the alert Smith. And, with time running out, Bridge-Wilkinson struck a superb free-kick which beat the Rovers' stopper but not the angle of bar and post.