Liverpool were restricted to few chances as a rigid Plymouth defence kept them at bay, teenage forwards Sheyi Ojo and Ben Woodburn wth their best opportunities in the first-half.
The Reds looked to first-teamers Daniel Sturridge, Adam Lallana and Roberto Firmino from the bench in the second-half, but none could make a telling impact against the Pilgrims, 68 places down the football pyramid.
The stalemate means Jürgen Klopp's team will have to travel to Home Park later this month - one in which they already faced a further five fixtures in 23 days, including two legs of an EFL Cup semi-final with Southampton, a crunch clash with Manchester United at Old Trafford and a heavyweight showdown with Chelsea.
Klopp makes history with youthful XI
Jürgen Klopp rang the changes for Liverpool, only Emre Can keeping his place from Monday's draw with Sunderland, and in doing so made history by naming the club's youngest ever starting line-up.
With an average age of 21 years 296 days, notable changes saw Joe Gomez handed his first appearance in 15 months after a nightmarish spell of injuries, while forward Ben Woodburn made his first start.
The 17-year-old Welshman, having scored in his only other senior appearance, an EFL Cup victory over Leeds United in November, was joined by fellow Academy prospects Ovie Ejaria and Trent Alexander-Arnold, making their third starts of the campaign.
In goal, Loris Karius came back in for his first start since the draw with West Ham United on 11 December, while winger Sheyi Ojo - who became the club's youngest ever FA Cup goalscorer in the third round of this competition last term - made his first appearance of the season.
With Lucas Leiva the oldest player in the line-up by almost seven years, Klopp's side easily surpassed the 52-year record set when Billy Shankly's team to face Wolves in the first division in 1965 averaged an age of 22 years and 303 days.
Plymouth meanwhile made just the one change despite the fact their last outing, away at Barnet, resulted in their first defeat in seven games.
Striker Craig Tanner dropped to the bench with defender Gary Miller coming in as Derek Adams fortified his back-line.
That meant that Oscar Threlkeld moved from left-back to central midfield, and allowed Liverpool supporter and the club's leading goalscorer this term - Graham Carey to take up an advanced midfield role behind a front three of Jake Jervis, Jordan Slew and on-loan frontman Paul Arnold Garita.
Ojo and Woodburn both go close
Liverpool's youthful exuberance saw them on top in the game's opening exchanges - Ejaria firing a sighter wide early on before Kevin Stewart, a year to the day he made his debut in an FA Cup draw with Exeter City, drilled well wide of Luke McCormick's goal.
As to be expected, the home side dominated the possession as they looked to open up spaces between the Pilgrims' well-drilled back-line - and Liverpool should have taken advantage of the visitors' inability to clear Can's free-kick.
The German midfielder saw his 25-yard set-piece blocked by a header but refused to give up the follow-up and after an aerial duel, the ball fell inadvertently for Ojo just a couple of yards from goal, but his header was far too tame to trouble the Plymouth 'keeper.
McCormick was on hand to deny another of the Reds' exciting teenage talents minutes later after Can had stormed in to win a loose ball and found Woodburn inside the box. The No.58 neatly turned his way past his marker, only for the 'keeper to spread himself well and deny his close-range poke.
Reds dominate possession but fail to create
Liverpool thought they'd found the breakthrough on 21 minutes when Divock Origi fired beyond McCormick and into the far bottom corner, but for the whistle of referee Paul Tierney to deny them for a perceived handling of Gary Miller.
And a disciplined Plymouth defence limited Liverpool to few goalscoring opportunities as they often everyone but Garita behind the ball, Stewart again firing harmlessly wide from the edge of the area.
Liverpool's wide players, Ojo and Woodburn, both showed neat touches as the half wore on and threatened to get in behind, but the Reds' end product continually disappointed. They produced just two shots on target from 11 total attempts in the first 45 minutes.
And despite their sheer dominance of the ball, seeing 80 per-cent of it, they were left frustrated by a resilient Plymouth team whose only meaningful attack came in first-half stoppage time and resulted in a harmless free-kick easily cleared.
Visitors enjoy good spell early in second-half
But while Plymouth showed little ambition to get forward in a first-half in which they completed only 92 passes, they stuck to their game-plan excellently and maintained a solid defensive shape.
Indeed it took Plymouth until the 50th minute to have their first shot, which their buoyant 8,500-strong away support immensely enjoyed although Slew's effort flew wildly wide.
Yet Adams' charges showed greater intent to get forward much more regularly than the first-half, bringing the first save out of Karius after 55 minutes - though he comfortably caught Carey's long-range free-kick.
And Liverpool still struggled at the opposite end to forge any real goalscoring opportunities, Lucas seeing a rare shot on target well saved despite getting good power behind it.
Plymouth hold on for shock draw
Klopp turned to Daniel Sturridge from the bench on the hour mark as he looked to inject some much-needed quality in the final third, the England international joining Origi up front.
The striker, having scored in his last two games and forced the last-gasp winner at Everton in his previous run-out before then, almost made a telling first contribution but his side-footed curled just wide.
But Liverpool continued to press for the game's opening goal and Sturridge was often involved, his cross just evading the stretched leg of Ejaria inside the six-yard box.
After a lengthy stoppage in play for Miller, who was stretchered off in serious discomfort, Klopp made two more changes as he looked for his side to avoid a replay in an already busy January - Adam Lallana and Roberto Firmino coming on for Ejaria and Woodburn.
That gave Liverpool four recognised forwards on the pitch, in addition to attacking midfielder Lallana, and as they pushed high up the field in their numbers, they were left vulnerable on the counter. On one such instance, they were indebted to Stewart's crucial tackle.
The midfielder had to step across to prevent substitute Craig Tanner from running through on goal after a fine first touch took him beyond Gomez and looked as though it would take him one-on-one with Karius in front of the Plymouth fans.
Liverpool desperately pressed for a decisive goal in the final 10 minutes, Sturridge heading over from Firmino's cross and then finding the side-netting at the near post after making space inside the box.
They could have stole it late on, but no-one was on hand to meet Jervis' cross, while McCormick dived low to prevent a deflection off substitute Connor Smith from breaking Plymouth hearts deep into stoppage time.
With that, on their first meeting with Liverpool since 1955, Plymouth held on to claim a huge draw and force a second meeting between the two teams later this month.