Struggling Farnborough stunned City when they took an early lead through Carl Gibbs before the Grecians pulled level with a Sean Devine strike.
Exeter left it late before Flack nodded home the winner to keep Exeter in the promotion-chasing pack.
City started brightly and Gareth Sheldon volleyed wide inside 40 seconds.
But it was Farnborough who took the lead in the 11th minute. Jay Smith picked out Gibbs inside the box and his powerful low shot from ten yards flew in off the post.
The goal stunned Exeter, but it took just ten minutes for the Grecians to pull level. Sheldon nodded the ball through to Devine who unleashed an unstoppable volley from 20 yards into the top corner.
The goal was no more than Exeter deserved after dominating much of the first half.
Devine almost gave Exeter the lead when twisted and turned into the box, but struck his shot just over the bar.
Farnborough had leaked 13 goals in their previous two Conference games, but Exeter still had difficulty breaking them down in the first period.
After the break Exeter continued to press forward and Sheldon struck a long-range free-kick inches over the bar.
Sheldon then forced Andy Petterson to save well before Flack headed the ball into the arms of the goalkeeper. Petterson again saved well to deny Les Afful as Exeter looked certain to stretch the scoreline.
Farnborough continued to frustrate Exeter, but the Grecians finally pulled ahead in the 77th minute. Sheldon's free-kick was floated into the box and Flack rose highest to nod the ball home at the far post.
In injury-time Jake Edwards should have stretched the margin of victory, but he fired wide from ten yards with only the goalkeeper to beat.