Brian Jensen proved there is life in the old dog yet as he produced more penalty heroics to take the Clarets into the third round of the Capital One Cup.
Starting for the first time this season, the 37-year-old made three superb saves to deny Plymouth and set the stage for Brian Stock to fire Burnley into the last 32.
Sam Vokes and Stevie Hewitt also kept their nerve to score from the spot but it was the long-serving Jensen who took the plaudits as he extended his 100 per cent record in shoot-outs to five.
His last one-man stand for the Clarets had come at Chelsea en route to the semi-finals of the competition in 2009.
And while this occasion lacked the same sense of drama, his contribution was just as important.
Needing a win to restore some momentum to their early season efforts, the Clarets had been taken to extra time and then penalties by a persistent Plymouth side.
Charlie Austin, the only player from Saturday's Championship defeat at Huddersfield to be given a start, had given Eddie Howe's re-shuffled side a first-half lead as he headed in his fourth goal of the season.
But a successful penalty in the final minute of normal time gave the underdogs a shot at glory and made sure Burnley had to fight all the way to reach the last 32 for the sixth year in a row.
Howe had used the opportunity to give game time to the players who hadn't figured much so far this campaign but probably didn't envisage most of them getting 120 minutes.
Stock and Michael Duff both lasted the distance, while the impressive Marvin Bartley made it into extra time but with so many changes it was perhaps inevitable that Burnley didn't click straight into gear.
While there was a willingness from Howe's experimental side to play possession football and pass their way up the pitch, a paucity of chances always kept the tie alive.
Plymouth were neat and tidy themselves but offered little threat on Jensen's goal, the nearest the visitors coming before half-time an early free-kick from Robbie Williams which comfortably cleared the bar.
A set-piece also brought Burnley's first serious attempt, although Jake Cole dealt easily enough with an effort from former Argyle loanee Alex MacDonald.
And it took the intervention of Austin eight minutes before the break for Burnley to take a grip.
The striker's selection provided something of an insurance policy for the Clarets and when Bartley produced a magnificent cross on the run, Austin arrived on cue to nod in and notch for the second round running.
Evidently not entirely satisfied, Howe replaced Keith Treacy with Ross Wallace at the break and ten minutes into the second half withdrew Austin following a harsh booking with Vokes handed the chance to get on the scoresheet.
But with just one goal still in it there was always hope for the visitors and although they struggled to get a clear sight of goal, a skidding drive just past the post from Williams reminded the Clarets that they still had work to do.
That effort gave Plymouth renewed belief and with Burnley rarely threatening to add to their lead - Vokes and MacDonald going closest with heading opportunities - the Pilgrims pushed late on to try and force extra time.
With the Clarets' back four standing firm there were few anxious moments but with a minute to go Plymouth's endeavours paid off as Vokes bundled over Paris Cowan-Hall and Williams calmly equalised from the spot.
That sparked a sense of urgency not previously seen in the home ranks but despite five minutes of added time they couldn't force a winner before the extra half hour with a shot on the turn from Vokes deflected over the top the nearest the Clarets came to avoiding the night shift.
Vokes then fired wide from distance as the 100-minute mark approached before Argyle substitute Onismor Bhasera headed into the side netting at the other end.
As the prospect of penalties loomed ever larger Bhasera was in the right place to divert a Danny Lafferty header over the top seven minutes into the second period of extra time and the shoot-out was confirmed when Jensen held onto a long-range drive from the dogged Conor Hourihane.
With Plymouth going first Williams stepped up again but this time Jensen - starting his testimonial season - guessed right to make the first of three top-notch saves.
With Ross Wallace seeing his effort saved and Junior Stanislas skying his attempt there was still work for the Clarets to do.
But Hewitt, who got better as the game went on in his first senior start, kept his cool to convert and with Matthew Lecointe and Maxime Blanchard doing likewise it was left to Stock to mark his full debut with the clincher.
Burnley: Jensen, O'Neill, Lafferty, Duff, Mee, Stock, Bartley (Stanislas 98) Hewitt, MacDonald, Austin (Vokes 55), Treacy (Wallace ht).
Subs: Trippier, McCann, Edgar, Grant.
Booked: Austin 50, Bartley 88
Plymouth: Cole, Nelson, Blanchard, Berry, Williams, Hourihane, Young, Gorman (Bhasera 67), Cowan-Hall, Chadwick (Lecointe 55), Sims (Lennox 67).
Subs: Feeney, Chenoweth, Richards, Gurrieri.
Booked: Hourihane 35, Lennox 76, Williams 108
Referee: Andrew Madley