Both sides scored one try each, but the boot of Wasps fly-half Alex King – he kicked one conversion, one penalty and one drop-goal – saw his side home by a narrow margin.
King opened his side's account with a seventh minute penalty from close-range after Saints were penalised at a scrum on Wasps' feed.
New Zealander Bruce Reihana eventually got his side on the scoreboard in the 25th minute, but by then the Powergen Finalists had already lost the services of their kingpin fly-half Paul Grayson, courtesy of ankle injury sustained in a heavy tackle from Rob Howley of all people.
A 38th minute King penalty put Wasps ahead again, but when the self-same King hacked a loose ball through just seconds later – from a stray Saints pass – centre Fraser Waters was on hand to dot the ball down for the first try of the game.
King made no mistake with the conversion and the home side held a 13-3 lead at the end of an error-ridden first half.
The second half had an eventful start with powerhouse wing Ben Cohen beating his England colleague Josh Lewsey at the left-hand corner for Saints' first five-pointer of the match, Reihana making no mistake to narrow the scoreline to 13-10.
King added his third penalty moments later when Saints front rower Mattie Stewart was penalised and sin-binned for killing the ball with Wasps on attack, but the scoreline remained at 16-10 for his 10-minute spell in the sin bin.
With just over 20 minutes remaining, and a possible place in the Zurich Final, both sides threw everything at each other, with Wasps landing the eventual killer-blow just on the 60-minute mark when King struck a well-timed drop-goal to stretch the scoreline to 19-10.
That nine-point lead proved to be enough to see Wasps home in the end, with the table-topping Gloucester outfit their next target and Fortress Twickenham the venue for the Premiership Final.
For London Wasps:Try: WatersCon: KingPens: King 3DG: King
For Northampton Saints:Try: CohenCon: ReihanaPen: Reihana