Cureton struck the first inside the opening five minutes as the Grecians started well. The veteran forward was in the right place to slide home from six yards from Daniel Nardiello's fine cross.
Exeter had chances to increase their lead, but were pegged back as Orient's Dean Cox equalised in the 13th minute in spectacular style, lashing the ball into the top corner from 25 yards.
Cox hit the post as the visitors enjoyed their best spell of the game just before half-time, but with 13 minutes remaining Cureton struck again, his 19th of the season, following up Richard Logan's saved shot to tap into an empty goal.
The goal took the stuffing out of Orient, who rarely tested Exeter goalkeeper Ben Hamer in the second half.
The victory takes Exeter level on points with Orient, five points adrift of the final play-off place with four games remaining.
Source: PA
Source: PA