Danish goalkeeper Brian Jensen kept Plymouth at bay to ensure Burnley remain the only Championship side to be unbeaten away from home. Jensen tipped over Tony Capaldi's shot and kept out an Akos Buzsaky volley and a Marcel Saip header in the first half.
He had to thwart Buzsaky again but it was the crossbar that rescued him when he was beaten by Barry Hayles' effort.
Andy Gray could have pinched it for Burnley but shot wide and was later denied by Argyle keeper Luke McCormick.
Plymouth boss Ian Holloway:
"We are building something here and although we didn't get the result today, we are progressing. "It was a strange old game because there was a swirling wind out there. The sides were very evenly matched.
"It was not the prettiest of games, but I was quite pleased with the look of my team," he added.
Burnley boss Steve Cotterill:
"I thought we might have nicked it at the end but that would not have been a fair result really." Cotterill also revealed his concern for striker Gifton Noel-Williams, who limped out of the game after 28 minutes with a groin injury.
"We will know how he is on Monday," said Cotterill.
Plymouth: McCormick, Connolly, Seip, Aljofree, Capaldi, Norris, Wotton, Nalis, Buzsaky, Hayles, Ebanks-Blake (Samba 86).
Subs Not Used: Larrieu, Doumbe, Summerfield, Djordjic.
Booked: Hayles, Wotton.
Burnley: Jensen, Duff, Foster, Sinclair, Harley, Jones, Hyde, James O'Connor, Elliott (Mahon 81), Gray, Noel-Williams (Lafferty 27).
Subs Not Used: Branch, McCann, Spicer.
Booked: Sinclair, Duff.
Att: 12,817
Ref: P Miller (Bedfordshire).
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