The legendary former England international saw his Nationwide Conference North side eventually outclassed by Graham Westley's side in the second period, after a highly competitive first half.
No doubt Gascoigne will have appreciated the skills of Boro's highly-rated 20-year old George Boyd, who proved the difference between the two sides, creating the first and lashing home the second for the visitors.
Kettering had threatened to give Gascoigne a dream start in the third minute when Craig McIllain's powerful header from a Jamie Patterson corner was clutched under the crossbar by Alan Julian.
But the visitors went ahead in the eighth minute when Boyd skipped around Wayne Diuk down the left touchline and delivered a pin-point far post cross which was met by the head of Darryn Stamp and the ball was perfectly placed back across goal and past Mark Osborn.
Kettering quickly hit back an equalised in the 15th minute when James Gould's cross was only half cleared into the path of Neil Midgley and the former Canvey Island striker hit the target from 12 yards.
Kettering had the better of the chances for most of the half with headers from McIllain and Midgley, both just missing the target, but two minutes before the break Stevenage regained their lead through the excellent Boyd.
The midfield man picked up the ball 25 yards from goal and advanced before firing a fierce left-foot shot which was deflected into the top corner past Julian.
The second half was a pale imitation as Kettering struggled to mount any sort of serious threat and Boyd could have notched his second with a rasping shot saved by Osborn in the 62nd minute.
The final stages were dominated by Stevenage and substitute Anthony Elding made it 1-3 with 20 minutes left when he stretched to convert a Dannie Bulman cross, before three times going close to adding to the Stevenage lead from 15 yards.