Evans booted over 16 points from six attempts and has now scored from 13 consecutive kicks at goal in two games to start the season.
Both teams won their opening game last weekend which meant the victors topping the Premiership, if only for 24 hours.
Despite losing openside Will Skinner shortly before kick-off - he complained of a dizzy spell having suffered a head injury last weekend - the hosts struck first.
Quins drove at pace through 10 phases from kick-off before being halted on the Saints' line.
But referee JP Doyle had signalled for an earlier offence and Evans stroked over the fourth-minute penalty.
In less than a minute, Quins were back on the front-foot, pinning Northampton back in their half with at considerable pace.
When Saints did get possession, Brian Mujati's drive was turned over in the tackle then Stephen Myler's kick to touch went out on the full.
Saints earned a penalty on 10 minutes - Joe Marler the offender in a maul - and they kicked to touch with more accuracy. But the lineout was undone and Quins stormed straight back upfield.
It was noticeable that Harlequins were playing at a quicker pace than Saints during the first quarter, the visitors shipping the ball across the line and back without penetrating the home defence.
Saints did recover from their early scrambles, however, and gradually established a foothold in what was developing into a tight battle.
But it was Quins who extended their lead in the 20th minute. Two Saints went off their feet defending their line and Evans made no mistake with the resultant penalty.
Yet Northampton were ahead three minutes later. They kicked a penalty to touch, won the lineout, retained ball through the phases in which Noah Cato was key, and Greg Tonks smashed through two tackles to score from close-range on the right.
Myler converted to make it 7-6 but the lead lasted only six minutes.
Doyle spotted an offside as Quins bore down on the 22 and the deadly Evans, who kicked 19 points against London Irish at Twickenham last Saturday, nudged his team back in front.
Saints were now back under pressure and when Phil Dowson caught Mike Brown with a swinging arm, Doyle gave him a warning and Evans another penalty, which he dispatched without fuss.
Saints broke clear after some desperate defence and when Mujati won a penalty after a powerful drive in a scrum, Myler again kicked for touch. That led to a scruffy bout of juggling and another Saints scrum.
This time Quins did better, Martin Roberts was sacked trying to break from the back and conceded a penalty for not releasing.
Quins began the second half like they did the first, only this time they scored a try.
Karl Dickson broke off the back of a scrum, dummied one challenge and armed hooker Joe Gray to power through by the posts. Evans added the conversion.
Myler kicked a goal to halt the scoreboard run, but the relief was temporary as Quins scored their second try off a move that had fans out of their seats.
Sublime back-play seemed doomed to failure confronted by a solid defence - but Evans and Jordan Turner-Hall executed a lovely loop-switch that released Ugo Monye to score on the left.
The one area where Saints were dominant, however, was the scrum and they won another penalty for Quins collapsing which Ryan Lamb scored.
Instead of taking control with a substantial lead, Quins began to display serious jitters.
It got worse when James Johnson was sin-binned six minutes from time for going off his feet to halt a Saints drive.
But Quins managed to shut out a side who failed to turn huge possession in the final quarter into points.
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