The Scarlets started at a startling pace, as from first-minute possession on the halfway line the ball was spread across the three-quarters to Stoddart, who burst a tackle before showing the pace to make it to the line.
The home pack recovered possession and encamped on the Quins line, the ball eventually coming out to Jones who was just held out on the left.
Stoddart failed to take a difficult pass in the Quins 22 that would have sent him under the posts, before a further gilt-edged opportunity presented itself to the home side.
The ball was spread across the back-line with a huge overlap awaiting the eager Jones, but the final pass was behind the wing, and after gathering he barely sneaked over the line, his try finally awarded by the television match official.
Man-of-the-match Malone replied with a penalty for the English visitors, before Stephen Jones atoned for his two conversion failures with a brace of penalties for the Scarlets.
Darren Daniel, a late replacement for the injured Dafydd James, looked like flying in after pouncing on a Stephen Jones clearance, but Quins full-back Mike Brown hauled him down five metres from the line.
A Scarlets penalty from the resultant position was slotted by Stephen Jones to take the half-time advantage to 16 points.
Malone quickly reduced the deficit with two second-half penalties, before Stephen Jones replied with his fourth successful kick.
With half-time replacements Tani Fuga, Nick Easter and George Robson thriving the Quins were a rejuvenated side, their pack getting on top as they drove towards the Scarlets line.
The Stradey Park team conceded a penalty, and Care showed the awareness to take a quick tap 10 metres out, catching the defence napping as he secured his side's first try.
To compound Scarlets misery, hooker Rees was sin-binned for a dangerous tackle on replacement Tani Fuga, Malone landing the penalty to reduce the gap to three points.
Mark Jones just managed to ground his first-half try
Malone then struck a dagger blow to the home side, a chip over the top of the defence in centre field catching Daniel cold.
Monye raced past the young wing, gathered the ball and cruised in to give his side the lead for the first time in the game.
The desperate Scarlets' challenge petered out, Malone sealing the win with a last-minute penalty that still left the home side the very slim consolation of a losing bonus point.
The Scarlets - who have one EDF Energy Cup game at Stradey before they say farewell to their famous home - have now lost their last eight Heineken Cup games in succession.
The Llanelli side face a testing trip to Stade Francais next week, while Quins host Ulster.
Scarlets coach Nigel Davies: "We are all bitterly disappointed, there is a lot of emotion in the dressing room.
"We felt we should have had the game won at half-time, we could have scored another three or four tries in that first half.
"But everything we talked about at half-time we didn't do in the second half.
"We didn't manage the game, we didn't play territory and we tried to play from silly errors. I thought we had what we deserved in the end.
"We can play rugby, there is no doubt about that, but when it comes to the big games we have got to learn how to win them."
Harlequins coach Dean Richards: "We knew at half-time we could get back into it. They could never play with the pace they did in that first half for 80 minutes, while we we couldn't play any worse than we did in that first half.
"We kept the scoreboard ticking over and we took our tries well. We controlled the game in the second half and played some great rugby.
"The boys deserve a lot of credit for that, but we know it is still very much early days in this competition and we are not going to rest on our laurels.
"The three boys who came on midway through the half [Fuga, Easter and Robson] also made a big impact for us."
Scarlets: Scarlets: Stoddart, Daniel, King, G. Evans, M. Jones, S. Jones, Martens, I. Thomas, Rees, Meeuws, Cooper, Maling, Easterby, G. Thomas, Lyons. Replacements: Higgitt for King (67), C. Thomas for M. Jones (74), Roberts for Martens (61), N. Thomas for Cooper (71), MacLeod for Maling (25). Not Used: Owen, John.
Sin Bin: Rees (59).
Harlequins: Brown, Amesbury, Tiesi, Turner-Hall, Monye, Malone, Care, Jones, Botha, Ross, Kohn, J. Evans, Robshaw, Skinner, Guest. Replacements: Gomarsall for Care (73), Fuga for Botha (55), Robson for Kohn (55), Easter for Guest (55). Not Used: Lambert, Taione, Luveniyali.
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