It was the Saints' fourth defeat in a row, but they looked to be heading for victory at Franklin's Gardens after coming back from 21-6 down to lead 30-24 with the clock dead.
But on his Saints debut prop Ethan Waller was yellow-carded for killing the ball in his 22 and Quins tapped it for Botica to barge over under the posts. The Kiwi knocked over the simple conversion to secure a win that they could scarcely believe.
Northampton dominated the scrum all game winning plenty of penalties from it, but they gifted Quins three first-half tries and ultimately paid the price.
Ryan Lamb started the scoring after three minutes with a penalty, but a wild pass from the fly-half set up the visitors for the opening try. Quins moved the ball to wing Sam Smith who stepped inside the disorganised Saints defence to scoot under the posts from halfway. Botica converted to give the Premiership champions a 7-3 lead after nine minutes.
Northampton's second-choice front row, with Paul Doran Jones at tighthead, drove a scrum seven metres to win a penalty and Lamb reduced the arrears to one point.
The home side had plenty of territory, but they could not take advantage and instead Quins profited from another wayward throw. This time Mike Haywood's lineout throw 10 metres from his try-line over-shot and Harlequins scooped up the loose ball for centre Harry Sloan to score under the posts. Botica made it 14-6 after 28 minutes.
Lamb hit the post with a 42-metre penalty, but Quins were gifted a third try from another poor pass from the Saints fly-half, which was intercepted by Smith on halfway to race under the posts.
Saints again came back putting pressure on Quins in the red zone, but they had to rely again on their scrum to gain three more points as they won a penalty in front of the posts.
Quins went in at half-time 21-9 in front with Northampton still searching for their first try for nearly three-and-a-half hours.
The home scrum again got Northampton going after the break, powering out a penalty for Lamb to add another three points, and from the re-start Saints finally broke their try-drought through Luther Burrell.
Jamie Elliott picked up the bouncing ball and his pass to Tom May saw the veteran full-back sprint clear of a tap tackle and pass to Burrell 20 metres out. The former Leeds centre held off two and slid over in the left corner.
Lamb missed the conversion and Botica immediately made it 24-17 with a penalty, but the home side had now got a taste for tries. More good work by Burrell saw him send over May in the same corner and from the same touchline Lamb converted to draw Northampton level.
The Northampton pack won their umpteenth scrum penalty and Lamb obliged seven metres in from the touchline to put Jim Mallinder's team in front for the first time with 15 minutes left.
Northampton scrum-half Martin Roberts was held up over the line and with Lamb on the upside of his rollercoaster performance he landed a drop goal with three minutes to go.
But with 42 second left Lamb kicked clear instead of holding on to the ball and when Waller was sin-binned with time up Quins and Botica took advantage.
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