Currie curled home a sublime effort in the 36th minute against the run of play to give Town a half-time lead. And after ex-Hatters' hero Danny Wilson's side squandered further opportunities, Goodall pounced six minutes from time from a Currie corner.
Hartlepool struck back in stoppage time with a Richie Barker penalty, but Kevin Blackwell's men held on to secure the points. Hatters handed full debuts to no less than seven players - Goodall, Richard Jackson, Chris Perry, Currie, Dave Edwards, Paul Peschisolido and Paul Furlong - as Town reflected the summer of change at Kenilworth Road.
Only Dean Brill, Chris Coyne and Matthew Spring remained from the team that lost 5-0 to Sunderland on the final game of last season.
Hartlepool fielded Robbie Elliott and Ian Moore, both signed from Leeds in the summer and also included Liverpool loan man Godwin Antwi in the heart of the defence.
Hatters started off keen to play the ball forward towards former QPR man Furlong at every opportunity.
It was the visitors who were more intent to keep the ball on the ground and just five minutes in James Brown latched on to Elliott's chipped pass and took the ball around Brill, but Luton escaped with just a corner as the winger ran out of angle.
Five minutes later Barker's flick from Willie Boland's pass released Moore in the area, but again the Pools striker was forced wide and could only fire an effort across the face of goal.
Hartlepool kept pressing and Boland slickly nipped between Edwards and Steve Robinson in the 17th minute to again play in Brown, running in from the right-flank, but Brill was quickly off his line to smother at the player's feet.
The visitors were dominant and should have opened the scoring in the 21st minute when, after head tennis in the area from an Elliott corner, Moore latched on to Goodall's short header back, but could only nod wide from five yards with just Brill to beat.
Luton finally woke up and Edwards volleyed a Spring cross wide from a tight angle in the 24th minute before Furlong headed a Jackson centre off-target two minutes later.
Town then took the lead in the 36th minute when Goodall's crunching tackle on the halfway line released Currie in the middle of the park and the former Ipswich man sidestepped his marker before curling a 25-yard, left-foot effort into the bottom corner of Arran Lee-Barrett's goal.
Luton were lifted by the strike, however it was Hartlepool who ended the half the stronger as Ritchie Humphreys smashed two efforts high and wide from good positions on the edge of the area in injury time.
Humphreys should have fired the equaliser just 20 seconds into the second half when Jamie McCunnie's cross from the right was flicked on by Barker to Humphreys, who had time to tee himself up before slamming a left-foot shot past the far post from 14 yards.
Goodall saw a 20-yard effort saved by Lee-Barrett in the 55th minute after he was teed up by Currie.
It was Goodall who had the next decisive act in the 86th minute when he headed home Currie's corner past Elliott on the line.
Luton were made to sweat late on when Barker smashed home a penalty in stoppage time, after Chris Perry had fouled substitute Anthony Sweeney.