Teenage keeper Cutts, making his senior home debut, blocked a fierce spot-kick from Reubus Richard-Noel in the first of the sudden-death kicks and Tyrone Fagan duly slotted home Town’s penalty to secure a second-round trip to Hitchin Town.
It was a dramatic way to end a tie full of entertaining play. Despite the shock of conceding after just 26 seconds, Carl Adams’ side then led 2-1 through Richard Gregory and Fagan, only to concede again from poor defending before half-time.
Liam Francis, Town’s skipper, looked to have won it with eight minutes to go but Poppies counterpart Steve Kinniburgh restored parity two minutes later.
After the lacklustre fare on Saturday in defeat to Frome Town, this was a much livelier, purposeful performance from Stratford.
Gregory, starting his first game since the opening day of the season, occupied a role behind Fagan and with Michael McGrath and Simeon Tulloch providing the invention, Town had plenty of enterprise.
They needed it too after conceding within 26 seconds. Richard-Noel made the incision down the right and when his cross was only partly dealt with, the ball fell to the right boot of Sandy and he smashed in a low shot past the helpless Cutts.
It was a test of character but Town responded. Fagan could have done better than nod wide from an inviting Tulloch cross and Tulloch’s curler drifted wide.
But the equaliser arrived on 18 minutes, Tulloch was again involved, crossing from the right and though Poppies keeper Craig Hill blocked’s Gregory’s first shot the striker scooped home the rebound.
Cutts did well to push aside a stinging 20-yarder from Andy Gooding but Stratford were ahead on 28 minutes.
McGrath fired in a free-kick from the halfway line and Fagan had time to send a pinpoint header from 14 yards into the top corner.
There was a real ebb and flow to the game. Back came Kettering and it was 2-2 on 36 minutes.
Town hesitated on the left as the Poppies pushed forward, Silvano Obeng snaffled possession before Richard-Noel’s flick sent Baker-Richardson towards goal. As Cutts advanced, the striker coolly slid the shot through his legs.
Stratford continued to create openings, Fagan having an effort smothered by Hill and Tulloch firing wide just before the break.
The second half continued in the same manner with Hill deflecting a low shot from Edwin Ahenkorah after the Town top scorer had come on for the injured Emmitt Delfouneso.
At the other end, Cutts was tested by a Saul Williams’ drive which brought a diving save.
Stratford looked the more likely scorers, though, and they regained the lead on 82 minutes, Francis soaring above the Poppies defence to send a thumping header from Gregory’s corner into the roof of the net.
Two minutes later, though, it was 3-3. A free-kick bobbled inside the area and Baker-Richardson flicked on for Kinniburgh slammed home.
And so to penalties.Elliot Sandy hit the bar with the Poppies’ first effort and Town left-back Kieren Westwood fired wide with Town’s opener.
All the other penalties struck the back of the net – Aaron Moses-Garvey, Gregory, Ahenkorah and McGrath for Town – to leave it 4-4 and into sudden-death.
Up stepped Richard-Noel but Cutts blocked his fierce effort and so Fagan had the chance to slot home the winner… which he did with aplomb.
TOWN: 4-2-3-1 Cutts; Delfouneso (Ahenkorah, 54), Francis, Simpson, Westwood; McGrath, Moses-Garvey; Tulloch, Gregory, Sheldon (Faulkner 79); Fagan. Subs not used: Kemp, Thompson-Brown, Evans.
KETTERING: 4-4-2 Hill; Harmon (Baker 72), Garcia-Krailing, Sweeney, Kinniburgh; Richard-Noel, Gooding, Sandy, Williams; Baker-Richardson (O‘Neill 89), Obeng. Subs not used: Djujic, Moussi, Archer.
Referee: Neil Radford.