Yorkshire resumed at 32-2, still 236 behind after centuries by Rehan Ahmed and Nick Kelly had put Leicestershire in a commanding position on day two.
They lost a third wicket inside three overs as wicketkeeper Ben Cox took a superb catch, diving to his right to remove Sam Whiteman via an inside edge, giving Davey a third wicket.
James Wharton and Matthew Revis added 31, but just as it seemed they might have found a way through a tough period, Alex Green entered the attack to take two key wickets in the space of four deliveries.
Generating good pace at the Bennett End, the England Under-19 international had Revis caught behind, driving, before bowling Bairstow through the gate, perhaps via a thin inside edge.
It was impressive stuff from the teenager, who stands at 6ft 6ins (198 cm) with broad shoulders to boot. Play was held up when he rapped Wharton on the right hand, the batter requiring lengthy treatment on the field before he could continue.
In the event, Wharton was the next to depart, stretching forward to defend against left-arm spinner Patel but somehow leaving a gap through which he was bowled.
Some calculated and profitable risk-taking by George Hill and Bess forced Leicestershire skipper Ben Green to take his namesake out of the attack, after which the seventh-wicket pair dealt well enough with spin at both ends to make it to lunch at 121-6.
Yet that changed with the first ball of the afternoon, Hill departing leg before to Patel, his front pad not far enough forward to be given the benefit of the doubt.
New man Hassan Ali was not long following him in, the Pakistan quick sending one skywards off Ben Green to be caught at short mid-wicket.
Bess and Moriarty held up Leicestershire’s push for victory for more than three quarters of an hour, but after 70 balls at the crease, Bess slashed at one from Patel outside off stump and was well held by Rishi Patel at slip.
Yet Moriarty excelled himself in completing a maiden half-century off 84 balls, hammering Patel down the ground for his 11th boundary, the last wicket adding 67 before Alex Green finally castled White to finish the job.
Report by ECB Reporters’ Network, supported by Rothesay.