Sutton's second-leg brace and Jamie Murphy's effort ensured there would be no European humiliation as the Scottish Premier League outfit turned around a one-goal first-leg deficit to clinch a deserved win at the Parc Y Scarlets.
Motherwell, who will now face Albanian side KS Flamurtari, came through a tricky test as they brushed aside their part-time hosts.
Jim Gannon's new-look side fell to a surprise home defeat north of the border last week, but they ensured there would be no upset here thanks to striker Sutton's early double and Murphy's clincher.
Sutton, who had already gone close with a diving header, made no mistake when he leapt highest to meet Steven Hammell's cross and cancel out the home side's one-goal advantage in the seventh minute.
It was one-way traffic as Steve McGarry headed over just over in a dominant start from the visitors.
And it was no surprise to see Well extend their advantage an claim a second vital away goal the 28th minute.
In a carbon copy of the first goal, it was giant striker Sutton who rose above his marker to head Paul Slane's cross beyond Llanelli goalkeeper Ashley Morris in the 25th minute.
Yet to their credit, Llanelli - the part-time side in Welsh Premiership - refused to throw in the towel as they searched for a way back into the game.
Stuart Jones, the matchwinner from the first-leg, twice came close to handing his side a lifeline.
The defender could chip over the crossbar after visiting goalkeeper Michael Fraser fumbled a free-kick. And Jones' frustrations continued when he headed player-boss Andy Legg's corner inches wide.
But it was Motherwell who still looked the most likely to score again as McGarry blazed just over from 25-yards out to leave Llanelli facing an uphill task at half-time.
The west Wales side began second-half on top as they searched for a way back into the tie.
Andrew Mumford almost gave the home side a glimmer of hope moments after the break with a header which was brilliantly tipped over by Fraser.
Mumford came close with a free-kick but the home side's purple patch failed to materialise in the vital goal.
And once the home side saw a handball shout turned down, there appeared no way back.
Motherwell made sure the tie was over as Llanelli, who threw men forward at will, were hit on the counter-attack as substitute Murphy killed off the tie with another headed goal.
Jim O'Brien skipped past Michael Howard before delivering the perfect cross for an unmarked Murphy to power home in the 72nd minute to ensure Motherwell's 500-strong travelling support went home happy.
It was the end of the road for Llanelli, who had dubbed the second-leg tie as the biggest game in their history.
But for Motherwell, their European adventure was just beginning as they breathed a sigh of relief at avoiding an upset and looked ahead to the next round.