The Premier League newboys were on course for their first win of the season through Gustavo Hamer's goal but Spurs sealed a dramatic victory with strikes from Richarlison and Dejan Kulusevski in the 98th and 100th minute respectively.
Following the game, Heckingbottom was furious at the officials for a number of incidents throughout the game.
"The officiating is appalling," he told Sky Sports in his postmatch interview. "It's not about the football decisions, just game management. My worry is that all the focus is on the yellow cards, time-wasting, the new guidelines.
"The focus all of a sudden comes onto time-wasting, so they're dictating how we play. If you play to take goal kicks from the back, you set up one way, and if Spurs go against that, that dictates the decision you make.
"We're getting told to just play long -- you can't do it. Wes [Foderingham] got booked for handling outside the box and then you're threatening him with another yellow [for dissent], you just can't do it.
"When I speak to the referees, they haven't got a clue what I'm talking about. They're officiating our games at the top level, and they simply do not know the game. We need to sort that, and sort it quick. It ruined the spectacle."
Heckingbottom was also upset that there was little protection for United goalkeeper Foderingham and questioned the decision to send off striker Ollie McBurnie late on.
"You see all the frustration," he added. "Both teams, both sides, both teams taking the mickey out of the ref and to sum it all up, we get Ollie McBurnie sent off for telling the referee someone's pulling his shirt.
"We've just seen someone turning their back, jumping into our goalkeeper, leading with an elbow, he needs stitches -- and that's deemed the same offence. What's going on with our game?"