Gareth Evans had given the Blues an early lead, only for Craig Mackail-Smith to cancel it out from the penalty spot.
It looked like the game was heading for a stalemate, until Tubbs popped up late on to send the large travelling support wild with delight.
Jayden Stockley came in as the lone striker in the Blues’ usual 4-2-3-1 system, with Tubbs dropping to the bench.
The visitors enjoyed plenty of possession in the opening stages, without being able to carve out a decent opening.
But they broke the deadlock with their first proper chance on eight minutes – although there was a slice of fortune involved.
Enda Stevens’ cross was cleared by former Pompey trainee Luke Wilkinson, but the ball ricocheted straight into his team-mate Scott Cuthbert and fell kindly for EVANS to slam home.
The lead did not last long, however, as Nigel Atangana barged into Josh McQuoid in the box and referee Peter Bankes pointed straight to the spot.
Mackail-Smith stepped up to take the penalty and confidently slotted the ball into the bottom corner.
After that frenetic opening, the game suddenly fell a bit flat, with the Blues unable to find the attacking fluidity that had been so evident in previous games.
Kyle Bennett did look dangerous when he burst down the left on 21 minutes, but Elliot Justham was quickly off his line to prevent Stockley from making a connection.
They soon received a let-off at the other end, though, when a poor touch from Christian Burgess forced Brian Murphy to exit his area and head clear.
Ryan Hall picked up the loose ball and surged into the box, only for Pompey’s keeper to make a quick recovery and smother the danger.
Bankes would then have breathed a sigh of relief on 34 minutes when he inadvertently intercepted Evans’ pass, but Cameron McGeehan fired wide to spare the official’s blushes.
The game suddenly livened up just before the break and Justham held a curling effort from Bennett before Mackail-Smith headed a Stephen O’Donnell cross wide.
Justham then did well to deny Stockley from close range, but had a simpler task in keeping out Gary Roberts’ angled shot.
Luton Town 1
It was the hosts who started the second half brighter, although it took until 54 minutes for them to create a serious scoring opportunity.
Pompey would have been responsible for their own downfall, however, after a mix-up between Burgess and his defensive partner Matt Clarke.
Mackail-Smith picked up the loose ball as Luton bore down on goal, but his pass for McGeehan was not quite right and the midfielder fired wide.
The Blues soon carved out a chance of their own as Stockley bent in a dangerous low cross that was just out of reach for the onrushing Evans.
Evans then met a Bennett corner with a glancing header that was heading for the bottom corner until a well-placed Olly Lee cleared it from the line.
Both sides looked capable of grabbing a winner, although it was Pompey who had the next opportunity when Evans was crudely felled by Hall.
Roberts stepped up to take the free-kick from 25 yards out and his effort struck the top of the wall and looped narrowly over the bar.
Cook made his first substitution on 73 minutes as Stockley made way for Tubbs, with Conor Chaplin replacing Bennett moments later.
Both sides continued to show plenty of attacking intent, although some resolute defending meant the goalkeepers were not being overly tested.
But there was still time for a late winner as Chaplin latched on to a long ball and held off a defender to get a shot away.
Justham stuck out an arm to palm clear the teenager’s effort, but TUBBS was on hand to convert the loose ball.
Luton (4-1-4-1): Justham; O’Donnell, Cuthbert (c), Wilkinson, Griffiths; O’Brien (Lee 46); McQuoid, McGeehan, Smith, Hall (Ruddock 82); Mackail-Smith (Marriott 71)
Goals: Mackail-Smith 15 (pen)
Booked: Smith, McGeehan, Wilkinson
Subs not used: Tyler, McNulty, McCourt, Green
Pompey (4-2-3-1): Murphy; Davies, Burgess, Clarke, Stevens; Atangana, Doyle (c); Evans, Roberts (McGurk 82), Bennett (Chaplin 75); Stockley (Tubbs 73)
Goals: Evans 8, Tubbs 90
Subs not used: Jones, Webster, Haunstrup, Hollands
Referee: Peter Bankes
Attendance: 9,083 (1,032 Pompey fans)