A first-half strike from Kwame Ampadu - his first league goal for more than six years - put Exeter ahead.
The Grecians were comfortable all night and were never really troubled by lack-lustre Dagenham. But with seconds left substitute Boot nodded home an equaliser at the far post.
Exeter started brightly and deservedly took the lead in the 11th minute. Alex Jeannin found Lee Phillips inside the box and when the ball broke to Ampadu he drilled the ball into the top corner from ten yards.
Jeannin struck a volley over the bar for Exeter as they continued to dominate the early stages.
Dagenham offered little as an attacking force in the opening period. Exeter always looked likely to stretch their lead and goalkeeper Tony Roberts saved well when Gareth Sheldon struck a powerful shot from the edge of the box.
Paul Buckle then headed the ball into the arms of the goalkeeper from Scott Hiley's cross.
On the stroke of half time City should have made it two when Jeannin's perfect cross was met by Phillips but his close-range header flew over the bar.
After the break Sheldon miscued a shot from ten yards as Exeter continued to push forward.
Dagenham attempted to take the game to Exeter but they always lacked the cutting edge in the final third.
Their best moment came when Chris Moore burst into the box and forced Paul Jones to make a save with a low 15-yard shot.
In the closing stages Andy Taylor struck a free-kick from the edge of the penalty area into the arms of the goalkeeper and Phillips struck a close-range shot over the bar.
But just when Exeter thought they had done enough to win the game, Scott Griffiths sent in a deep cross and Boot headed home for the visitors.