Potts delivered an outstanding spell with the old ball to remove Hassan Azad to allow the hosts to break into the Leicestershire batting order.
The 23-year-old found life out of the pitch that no other bowler was able to conjure, earning figures of six for 58 and passing fifty first-class wickets in the process.
Leicestershire's Colin Ackermann said: "We set out with the ambition of trying to bat out the whole day. Credit to Durham they bowled pretty well.
"Matthew Potts got fired up and bowled extremely well against us. It was disappointing not to bat until at least tea, but credit to them.
"Hassan Azad had a brilliant innings against Worcestershire last week, and hopefully he can replicate that again on Sunday.
"The equation is pretty simple, we have to set out our stall and bat for as long as possible and keep all the good balls out.
"I was very surprised that Durham didn't declare late on Saturday."
Ackermann added: "If I was in their position I would have tried to get 400 runs as quick as possible and had a go on Saturday evening.
"We were expecting a declaration, but obviously it didn't hand.
"Beuran Hendricks got struck while he was batting, but fortunately there is no fracture.
"We'll try our best to save the game, if a few batters get in we have to be ruthless and not give them a sniff."
The North-East outfit bowled out the visitors for 273 before building a dominant lead of 394 runs courtesy of a maiden first-class hundred from Michael Jones and a fine knock of 84 from Sean Dickson, who capitalised on an ailing Leicestershire bowling line-up.
A declaration will likely occur overnight, leaving the hosts requiring 10 wickets on the final day to force their first win of the campaign.
Durham were in need of inspiration with the old ball at the start of day three with Azad and Colin Ackermann seemingly entrenched, and Potts responded by bounding in from the Finchale End and hitting the wicket hard to prise out Azad, ending the opener's stubborn knock after 206 deliveries for 66 and a fine partnership with Ackermann for the third wicket of 112.
Azad was cramped up by the accuracy of Potts, which began a clatter of wickets in the Leicestershire middle order courtesy of the Durham seamer.
Louis Kimber lasted only three balls before he played on with a loose drive before Nick Welch was pinned lbw for only two to give the home side an opportunity to ramp up the pressure with the arrival of the new ball. In a five-over burst with the old ball, Potts claimed figures of 3-15 to open up the Leicestershire ranks.
Ackermann appeared to have stymied the threat before Raine produced a teasing delivery that found the outside edge for 81.
The home side continued to chip away and Potts passed a milestone in his career with the next wicket to fall, securing his 50th first-class wicket by removing Ed Barnes with a Scott Borthwick-David Bedingham double act in the slip cordon.
After a break from the attack, Potts was called back into action to break another Leicestershire stand.
With his first ball of his second spell of the day, the 23-year-old removed Harry Swindells to notch his first wicket-haul in first-class cricket. Potts' pace and bounce proved to be too much for Will Davis to handle, presenting a simple caught and bowled chance.
Liam Trevaskis wrapped up the visitors' innings, handing Durham a lead of 155 to take into the second innings.
The home side were able to capitalise on the absence of both Chris Wright and Beuran Hendricks due to injury as openers Jones and Dickson were able to accelerate at the crease reaching a century stand in the 23rd over.
Dickson passed fifty for the second time of the match from only 66 balls, although he was given a life after being dropped on 40 by Welch off Ackermann.
Dickson's attempt to elevate the run rate cost him his wicket 16 shy of a second century of the game. However, Jones succeeded where his team-mate fell, reaching three figures for the first time of his career with a firm strike through the covers before he was out before the close for 108.