Both goals came in the first half, the striker breaking the deadlock with a stunning shot from the right hand side of the box.
Riis netted his second from the penalty spot in added-on time at the end of the first period after the impressive Ali McCann had been fouled by Sonny Bradley.
On Friday, 23-year-old Riis had signed a new contract which runs until the summer of 2025, so this was the perfect way to mark that faith put in him.
He now has 11 goals for the season, seven in the Championship and four in the Carabao Cup.
North End were worthy of their victory, overcoming a nervous spell after half-time to get on the front foot again.
Riis had a couple of chances to notch a hat-trick, with Scott Sinclair also denied by the keeper.
North End were in their usual 3-5-2, with Tom Barkhuizen operating at right wing-back - Sepp van den Berg moved inside into the back three next to Patrick Bauer and Andrew Hughes.
The opening stages of the first half were drab to say the least, it taking until the 17th minute for a chance at either end, Barkhuizen shooting over the bar from the edge of the box.
It was out of nothing that PNE took the lead, a tremendous piece of play from Riis.
Collecting Barkhuizen's pass on the side of the box, there looked nothing on other than a cross into the middle.
But the Dane steadied himself with a touch and then put his laces through a right-foot shot, the ball rocketing across goal into the far corner of the net.
The atmosphere livened up as the North End faithful made their views known to referee Dean Whitestone on a couple of questionable decisions.
In the 36th minute, Riis got behind the Luton backline and turned to lay the ball off to Sean Maguire whose shot from outside the box deflected off a defender and over the bar.
Luton came close to levelling matters when a free-kick into the box found Glen Rea at the far post. His downward header was blocked by Daniel Iversen with the PNE keeper saving Kal Naismith's follow-up.
Two minutes into first-half stoppage-time North End got their second goal from the penalty spot.
McCann's run down the right hand side of the box was halted by Sonny Bradley's sliding challenge, Mr Whitestone pointing to the spot after a few seconds of thinking time.
With regular penalty taker Daniel Johnson on the bench, Riis stepped up and tucked the spot kick low into the bottom corner, beating the dive of keeper Simon Sluga.
Luton made a double substitution at the interval, changing formation with it.
One of the subs, Admiral Muskwe, had a shot from the edge of the box two minutes into the second half which Iversen dived to his right to tip past the post.
Just before the hour, Maguire's pass played in Riis behind the Luton defence. Riis stumbled slightly and when he regained his footing, pulled a shot wide from just inside the box.
Two chances came PNE way in quick succession as they looked for a third goal.
In the 64th minute Josh Earl's cross from the left was kept in by Barkhuizen on the byline to the right of goal, his low shot blocked by Sluga at the post and behind for a corner.
Ben Whiteman delivered the resulting flag kick into the middle, Bauer meeting it with a header which Sluga dived to tip over the bar.
Riis had the chance to complete a hat-trick with a quarter of an hour remaining, Alan Browne having fed a pass down the right-wing for Barkhuizen for chase.
Barkhuizen lifted a cross into the box which Riis met by headed over the bar.
Riis should have had a penalty in the 85th minute when he got the wrong side of Lockyer who appeared to push him as he tried to stop him breaking into the box.
But the referee waved play on, much to the disquiet of the PNE faithful.
In the 88th minute, Riis' through ball played substitute Sinclair clear on goal but with two defenders closing him down he put his shot too close to Sluga who spread himself to blocvk.
As the game moved into stoppage-time, Iversen pulled off a stunning save to keep his clean sheet in tact.
Kal Naismith let fly with a shot from 25 yards which took a huge deflection off Bauer, Iversen somehow getting a hand to the ball to turn it behind.
PNE: Iversen, Van den Berg, Bauer, Hughes, Barkhuizen, Whiteman, McCann (Ledson 90), Browne, Earl, Maguire (Sinclair 76), Riis (Potts 90). Subs (not used): Cunningham, Johnson, Storey, Rudd.
Luton: Sluga, Lockyer, Bradley (Lansbury 46), Naismith, Bree, Rea (Muskwe 46), Mpanzu, Clark, Bell, Cornick (Onyedinma 70), Jerome. Subs (not used): Hlyton, Burke, Campbell Shea.
Referee: Dean Whitestone (Northamptionshire)
Attendance: 11,059 (989 away)