Sutton slotted home a spot-kick just before the hour mark but was ordered to re-take the penalty after a home player encroached into the 18-yard box.
The goal should have stood but Sutton's second attempt was saved and Flamurtari soon took the lead when Ermir Strati's half-volley from the edge of the area bounced in off the post.
Although the hosts had hit the woodwork twice, the penalty save appeared to lift them and they comfortably held on to their first-leg lead in the second qualifying round clash.
Sutton had earlier hit the bar from close range as Motherwell began the game well despite difficult circumstances.
The match was played in searing heat, with temperatures hitting 38 degrees Celsius just before kick-off, and Austrian referee Fritz Stuchlik took the unusual step of stopping the match for a two-minute water break in each half.
Motherwell had little time to adjust to the heat - they only arrived in Vlore two hours before the game following a three-hour drive from Tirana.
But Jim Gannon's men looked comfortable from the early stages, keeping the ball better than their hosts, who were playing their first European game since 1996 after winning the Albanian cup in May.
Motherwell were unchanged from the side that beat Llanelli 3-0 last week to overturn their home-leg loss and young midfielder Ross Forbes was dictating play.
Wingers Jim O'Brien and Paul Slane looked to have the measure of their direct opponents but Motherwell could not deliver a telling ball for most of the half.
Flamurtari created the first opportunity but defender Franc Velui scuffed Roberto Alvizh's deep corner wide from close range.
Well continued to look comfortable and Steven Hammell curled a 25-yard free-kick just over after Forbes had been harshly booked for taking it before the referee was ready.
They came so close to an away goal in the 42nd minute but Sutton stretched to head against the bar from three yards after Keith Lasley had lofted the ball to the back post following good build-up play.
Well brought on Jonathon Page at right-back for Slane, who looked to have succumbed to the heat, but the Albanians almost snatched the lead in injury-time.
Substitute Odise Roshi headed Roberto Alvizeh's corner off the top of the bar and Well were rescued by the post just after the restart when Devis Mema shot low past Michael Fraser after Hair Zeqiri had opened up the away defence.
The visitors settled and were given a great opportunity to take control of the tie when Bledion Guga pulled Sutton's shirt when Hammell lofted a free-kick into the area.
Sutton sent the goalkeeper the wrong way but, after Hammell had been booked for protesting against Stuchlik's re-take decision, the striker hit his second attempt in the same place and Shpetim Mocka dived low to turn the ball wide.
The home fans in the crowd of around 4,000 raised the noise levels and Strati sent them into raptures when he sent his half-volley across Fraser after the ball broke kindly following a throw-in.
Motherwell pushed forward but struggled to create chances and did not make the best use of several set-pieces.
The hosts looked dangerous on the break and sub Elham Galica could have done better than shoot straight at Fraser from 20 yards.
Well sub Jamie Murphy got a sight of goal in injury-time but his 20-yard drive was comfortably held by Mocka.
The Steelmen will need to show more conviction in the final third at Airdrie next Thursday but Gannon will be optimistic his side can turn around the tie.