Pinkney took control at Kenilworth Road just 24 hours before the game and could have been left wondering what he had let himself in for after a simply dreadful first half from the hosts.
Plymouth led 2-0 at the break through David Norris and Peter Halmosi, but the scoreline could have been even greater.
Mercifully, the Hatters improved after the break as substitute Stephen O'Leary netted, but despite some late huffing and puffing Town never really looked like sneaking a leveller and now have just one win in their last 19 league matches.
With Barnsley winning against Southend, only Hull's failure to beat Colchester means Luton are not yet mathematically down.
Hatters made two changes to the side that lost 3-2 at QPR on Easter Monday as Kevin Foley replaced Ahmet Brkovic, who was out with an eye injury, and fit-again Leon Barnett was preferred at centre-half to Markus Heikkinen.
Plymouth, who kept the same 11 that beat Leicester 3-0 in their last outing, started much the better in the sunshine and after Barry Hayles went close from an early Akos Buzsaky corner, the visitors took the lead in just the fourth minute.
Halmosi had all the time in the world on the left flank to cross into the area and Norris ghosted in from deep to loop a header into the top corner from 15 yards to leave Luton shocked.
Calvin Andrew nodded a David Bell free-kick harmlessly wide in the seventh minute, but it was all Plymouth and Marcel Seip mis-kicked in front of goal a minute later after another Buzsaky corner had caused problems in the home defence.
Goalscorer Norris then skipped past Foley with ease in the tenth minute before slipping in Slyvan Ebanks-Blake, only for Chris Coyne to make a timely intervention.
Luton were struggling to cope with the Pilgrims' set-pieces and two more corners in quick succession in the 17th and 18th minutes saw Dean Brill firstly save from Lilian Nalis' far-post header before Mathias Doumbe nodded over from eight yards.
Hatters finally got a foothold on proceedings and threatened in the 24th minute when Coyne's flick released Steve Robinson, but the Irishman's first touch was far too heavy and visiting goalkeeper Luke McCormick was able to smother at the midfielder's feet.
Instead it was Plymouth who came close to making it two in the 34th minute as Buzsaky was allowed to waltz unchallenged through the midfield and find Ebanks-Blake, who spun on the edge of the area and hit a left-foot shot that had Brill wrong-footed only to creep past the post.
The second did arrive six minutes later as Ebanks-Blake and Hayles linked up with great effect and Barnett's half-challenge could only divert the ball into the path of Halmosi, who had plenty of time to drill a low left-footed effort past Brill from 14 yards.
It could have been three a minute before the break as yet another Buzsaky corner caused more danger, but Hayles failed to convert at the far post.
Plymouth dictated again after the break and Brill had to be at full stretch to tip aside a low 30-yard drive from the lively Halmosi.
From Halmosi's resulting flag kick, Paul Connolly saw his near post effort hit the side netting.
Luton had been troubled by corners all afternoon, but it was from one of their own that they pulled a goal back in the 51st minute.
Coyne met Bell's deep kick from the left with a thumping header at the far post that was blocked on the line, only for O'Leary to blast home the rebound from close in for his first of the season.
Lifted by the strike, Hatters improved and Lewis Emanuel curled just wide from 20 yards in the 68th minute after a neatly worked free-kick routine involving Robinson and Bell.
Andrew then drilled a shot just past the same post in the 73rd minute after the pumped-up O'Leary had touched Coyne's flick-on into the striker's path, but that was the best Town could offer.