Shocking 911 tapes emerged today that suggest cops responding to the horrific 2007 Connecticut home-invasion murder case made a series of glaring mistakes that may have cost the victims their lives.

The tapes — revealed on the sixth anniversary of the heinous “In Cold Blood”-like murders of a doctor’s wife and their two young daughters — show how Cheshire cops were clearly skeptical of nurse Jennifer Hawke-Petit when she walked into a bank and told the teller she was being robbed.

The fated mom, 48, said her captor was waiting outside while his partner had her two young daughters tied up at home.

“Apparently, she came into the bank, she tried to get some money out,’’ doubting police Lt. James Fasano can be heard telling dispatcher Donald Miller after Hawke-Petit and kidnapper Steven Hayes drove off.

“One of the accounts was in her husband’s name, and then she says, ‘Well, my kids are at home tied up.’ So we don’t know if they really are or if she was just trying to get money out at this point,’’ Fasano said. on the call.

“She was calm. She didn’t appear upset,’’ he said.added the officer, Lt. James Fasano. “She walked into the bank, she got the money, she was by herself. The other person was in the car.’’

The tapes — obtained by the Hartford Courant — indicate that the clueless cops were even still holding onto the twisted notion that Hawke-Petit might be complicit in the bank scheme after they arrived at the family’s home.

“Three suspects. One is a female supposedly in the upstairs bedroom possibly [dead], with the other two,’’ said Cheshire Sgt. Robert Vignola Cheshire Sgt. Robert Vignola over the radio.

That was about 50 minutes after the initial 911 call.

Those errors appear to have led the police to take a disastrous hands-off approach to the developing scene.

One police officer, who was in the neighborhood when the initial 911 call came in at 9:21 a.m. July 23, 2007, actually got to the house before Hayes drove Hawke-Petit back to the house from the bank.

But he was ordered not to approach anyone as he watched the SUV pull into the garage — or even make a phone call inside to try to determine what was going on.

“What I want to do is set up a perimeter, a better perimeter, as soon as everyone is geared up, and then what we are going to wind up doing is … we’re going to have to make contact,’’ Sgt. Vignola instead told Lt. Joseph Mazzini.

At one point, a police hostage negotiator called in but was told to hold off, the tapes show.

Inside, Hayes, 44, and partner Joshua Komisarjevsky, 27, were brutally beating and tying up Hawke-Petit’s husband, endocrinologist Dr. William Petit, in the basement.

Hayes then raped and strangled Hawke-Petit, and Komisarjevsky sexually assaulted the couple’s youngest daughter Michaela, 11.

Both Michaela and sister Hayley, 17, were still tied to their beds and alive when the fiends set the house on fire. William Petit escaped and was the sole survivor.

The killers were caught as they tried to flee.

Hawke-Petit’s sister, Cindy Renn said she believes that the police “from Day 1 didn’t do what they should have done.”

“And they’ve never admitted it,’’ Renn told The Post. “It’s a huge cover-up, and I don’t know why. Why can’t you just say, ‘We really messed this up, and we’re wrong, and we’re sorry’ ?’’

“I don’t want to stir up anything wrong or bad. … But if a police force is never asked [to] reevaluate, then I don’t think anything is every learned.’’

The 911 tapes were obtained by the Courant from law-enforcement sources — after the Police Department said they had been destroyed in a freak lightening incident in September 2010.

The Courant said its copies were made after September 2010.

When reached at the Cheshire Police Department, Fasano told The Post no comment and said no one else there would be talking, eitherThe Cheshire Police Department told The Post it had no comment on any of the developments.

The tapes surfaced on the same day that the Hawke and Petit family and friends laid roses at the graves of their loved ones to mark yet another sad anniversary

Every year, “We have a Mass for the girls, and then we go to the cemetery to visit as a family. We just talk about the girls,’’ William Petit’s mom told The Post.

An HBO documentary called “The Cheshire Murders,’’ which aired a day earlier, questioned the police response in the case.

Both murderers are currently on Connecticut’s death row. Additional reporting by Laurel Babcock in Cheshire, Conn.

ksheehy@nypost.com