Quins flew into a 27-12 lead during a frantic first half, with Fraser Dingwall helping to set up both Saints tries, for Tom Collins and Tom James.
Saints came out of the blocks quickly in the second period and Sleightholme delivered an eye-catching breakaway score that started well inside his own half.
But Marcus Smith continued to stick the boot into the black, green and gold, who didn't make an abundance of possession count during the second period.
Quins started to concede an abundance of penalties in the final 20 minutes and found themselves down to 13 men.
However, Saints were profligate, failing to hold onto the ball and letting their hosts off the hook as they saw the game out to claim a victory and leave Chris Boyd's men with nothing.
It was an infuriating conclusion for Saints, who had the chance to claim at least two bonus points in the closing stages.
But they squandered their chances, with myriad inaccuracies costing them dear against a fellow top-four hopeful.
The day didn't start well for Saints as Matt Proctor (calf) and Piers Francis (dead leg) were forced to withdraw from the matchday squad.
That brought Collins and Rory Hutchinson into the starting 15, with Taqele Naiyaravoro on the bench.
And Harlequins were ahead inside two minutes as Nick Isiekwe, hampered by the sunlight, failed to gather the kick-off and the hosts won a penalty, which Smith kicked with ease.
Saints had a chance to respond after winning a penalty in Quins territory, but the kick missed touch and an opportunity to issue an immediate riposte drifted away.
However, it didn't take Saints too much longer to strike at the heart of Harlequins as they set up a move from the lineout and Sam Matavesi and Dingwall combined to send Collins over.
Hutchinson missed the conversion and he quickly lost the ball in his own half, allowing Quins hit back just a minute later as Danny Care put in a grubber kick and Tyrone Green dived on it to score.
Smith converted to make it 10-5 inside nine minutes and Quins scored again inside two minutes.
Sleightholme made a superb saving tackle but Quins kept the ball and kept it moving, allowing Stephan Lewies to stretch out an arm to score.
Smith made it 17-5 after an absolutely breathless opening 11 minutes, but Saints were straight back at the door, and scored a second try of their own.
Dingwall did brilliantly to scythe his way through in the build-up and James ran a superb support line to gather the perfect pass and score.
Hutchinson converted from a much easier position this time to cut the gap to five points.
The next 10 minutes strangely went by without a score, but Quins thought they had one when Care dotted down, only to be denied as Alex Dombrandt hadn't released in the build-up.
But Quins did strike after a big Dombrandt carry, with Care finding Mike Brown close to the line.
Smith converted to make it 24-12, while Saints were dealing with the loss of skipper Lewis Ludlam to a head injury assessment.
Smith added to the hosts' lead with a penalty two minutes before the break and Saints had a mountain to climb during the second period.
Ludlam was unable to return after the interval as he failed his head injury assessment, giving JJ Tonks a prolonged spell to impress in the Saints back row.
Saints came out with the bit between their teeth and George Furbank was just held up as his team-mates drove him towards the line.
Quins look like they would score on the counter-attack as they flew forward, but it was Saints who landed the sucker punch.
Dingwall gave the ball to Sleightholme and the speedy wing turned on the turbos from just outside his own 22, kicked ahead and gathered to score a sensational try.
Hutchinson converted and the deficit was down to eight points.
The try was Sleightholme's final act as Naiyaravoro was introduced, with the wing almost making an immediate impact, sprinting down the right before being tackled at the last.
Smith landed a penalty soon after to make it 30-19 in Quins' favour, but Collins and James threatened to respond as they linked up well before Smith intervened to give away a penalty.
At the other end, a Quins effort was ruled out for a forward pass in the build-up as the game continued to be played a lightning-quick speed.
Saints were still dominating possession, having 80 per cent of the share during the first 20 minutes of the second half.
Quins were shipping penalties at a regular rate, giving the black, green and gold good field position.
Finally a home player paid the price for the indiscipline as replacement flanker Jack Kenningham was sent to the sin bin, and Saints tried to turn the screw in his absence.
Quins lost Aaron Morris to a nasty-looking injury, which saw the game stopped for some time.
When the game resumed, another home player was hit with a yellow card as Dombrandt was given 10 minutes off the field, reducing Quins to 13 men.
Saints took the scrum but wasted the possession out wide, giving Harlequins a reprieve.
The away side were looking erratic and really struggling to make their numerical advantage tell.
Quins were getting let-off after let-off and with the clock ticking down, hopes of a Saints recovery were fading away.
Kenningham returned to boost the home ranks with five minutes to go and Saints still couldn't find a way through.
Instead, it was Quins who had the final say, twisting the knife with a breakaway score that ended with Luke Northmore setting up Brett Herron for a try.
Herron put the icing on the cake with the conversion as Saints headed home with nothing to show for their display.
Harlequins: Brown; Morris (Northmore 67), Marchant, Esterhuizen, Green; Smith (Herron 76), Care (Landajo 75); Marler (Els 63), Baldwin (Gray 67), Louw (Kerrod 56); Lamb, Lewies (c); J Chisholm, White (Kenningham 57), Dombrandt.
Saints: Freeman (Mallinder 56); Collins, Dingwall, Hutchinson, Sleightholme (Naiyaravoro 50); Furbank, James (Taylor 59); Auterac (Waller 50), Matavesi (Haywood 50), Hill (Painter 50); Coles, Ratuniyarawa; Isiekwe (Moon 60), Ludlam (c) (Tonks 27), Wood.
Referee: Luke Pearce