Reds boss Jurgen Klopp had made 11 changes and his side fell behind early on when Tom Nichols poked in.
Jerome Sinclair equalised with a low finish but the hosts retook the lead when goalkeeper Adam Bogdan conceded direct from a Lee Holmes corner.
With time running out, Brad Smith rescued Liverpool with a neat finish.
The replay at Anfield means Exeter are set for a cash windfall, with Sporting Intelligence estimating it will be worth about £700,000 to the Devon club.
Liverpool have been hit hard by injuries this season, with enough first team players on the sidelines to make up an alternative starting XI.
Consequently, Reds boss Klopp made wholesale changes for Friday’s game, with just three players boasting more than four first-team appearances.
A number of those included in the side had been recalled from loan spells elsewhere, including midfielder Kevin Stewart, whose spell at League One Swindon was ended just a few hours before the Exeter clash.
Stewart, defender Tiago Ilori and winger Ryan Kent were making their first-team debuts for the club, and the team’s lack of any significant time playing together was immediately evident.
They lacked cohesion, with Sinclair’s goal their only effort of note in a first half Exeter edged, and as the game wore on the home side grew in confidence.
Klopp’s gamble looked to have failed until Smith took advantage of a tiring Exeter to spare his manager’s blushes, but the German has previously complained about his side’s busy fixture schedule and the draw has added another game to that.
It has been a season to forget for Bogdan so far.
The Hungarian keeper made his Premier League debut against Watford on 20 December when number one Simon Mignolet was injured, but made the headlines for the wrong reasons when he dropped the ball from a corner to allow the Hornets to score.
Against Exeter, it was once again a corner that caught Bogdan out.
With the game approaching half-time, Holmes swung the ball in and it dipped under the crossbar as a perplexed, out-of-position Bogdan flapped meekly at it with an Exeter player in close attention.
Match of the Day co-commentator Danny Murphy was not impressed: “Unbelievable. The keeper has to be stronger – you can’t be bullied by a striker. You’re stronger than him and he’s weaker than you.”
Exeter are 16th in League Two and went into the game on the back of four successive defeats, but they clearly sensed the opportunity to cause an upset from the start.
They took the game to Liverpool, pushing their players forward at every opportunity and were good value for their ninth-minute lead.
Even after Liverpool’s equaliser Exeter did not alter their approach to their game. Instead it was their opponents who rarely progressed beyond the halfway line.
After Holmes caught out Bogdan to restore Exeter’s lead, it looked like the hosts were on course for a famous FA Cup upset.
Ultimately, their exertions took their toll as Smith took advantage of tiring legs, but a replay at Anfield is the least they deserved for a determined display.
What they said
Exeter boss Paul Tisdale: “I’m a football manager and I want to win games. When you’re 2-1 up and defending a lead with 10 minutes to go then you’re disappointed. But we got a draw against a very good side and have a replay at Anfield. We have no investor so every penny counts. At the end of the day we’ve put on a really good show.”
Liverpool manager Jurgen Klopp on BBC One: “It’s obvious what we have to learn in the future. We have to be more robust. It was a difficult pitch. Exeter did really well. Maybe they know which part of the pitch you can play football. It was really difficult for us today. On one-on-one situations they were more robust than our guys.
“The first goal was unlucky for us – it was their first attack. I won’t say anything about the second goal – they showed it live on the BBC. I can’t believe we have to play another game now. You want it like this, so we will be there.”
‘Liverpool youngsters will learn’
“It was a brilliant 94 minutes of football,” said BBC Sport pundit Alan Shearer. “From start to finish it was entertaining. What a night for the Exeter chairman and the football club.
“As for Liverpool, the experience the kids will get from the game will carry them on for the rest of their careers.”