The striker opened the scoring against the run of play on 18 minutes, giving the visitors a lead at the break.
Constable took his tally for the season to nine with another smart finish early in the second half and Peter Leven made it 3-0 from range two minutes later.
Jamie Cureton reduced the arrears with a hotly-debated penalty, but United held firm to see out the remaining 20 minutes.
It sealed United's best display on the road this season and extended their unbeaten run to eight games.
Lee Cox replaced the injured Adam Chapman in the only change to the starting line-up from the 2-0 win over Accrington eight days earlier.
Devon had been hardest hit by the wet weather over the last few days, but there were no issues with the St James' Park pitch.
The game began in strong winds and United, playing into them, took time to adjust.
Exeter looked to seize the initiative and looked very dangerous in a dominant opening 15 minutes.
Their main outlet was the veteran hitman Cureton, who forced Ryan Clarke into a smart parry from a free-kick on seven minutes.
When the ball came back in the 37-year-old showed a flash of brilliance with a swivelled volley which beat the U's goalkeeper but smacked off the crossbar.
The visitors were not doing themselves any favours, giving the ball away cheaply.
Alfie Potter's carelessness in possession gave Alan Gow a chance, but the midfielder delayed his shot and the chance disappeared.
Michael Raynes also needed to make a last-ditch block as United struggled to stay level.
But slowly they found a way to cope with the conditions and opened the scoring on 18 minutes.
Cox helped the ball into the box and clever work from Tom Craddock created space for Constable, who lashed the ball into the net from 12 yards.
It began a terrific spell for the U's, as City's three-man defence was overwhelmed.
Sean Rigg terrorised Scot Bennett, going past the converted midfielder every time he got the ball.
He laid on chances for both strikers in the space of a minute just before the half hour mark.
The first was fired over by Craddock from close range, while the second came at the end of a superb move. Leven played in the winger, whose centre saw Constable have a shot deflected just past the post.
Exeter steadied the ship and came back into the game, although their threat was not the same as the opening spell.
The closest they came was a free-kick from Gow, which flashed across the face of goal just past two City players.
United were forced into a substitution when Liam Davis appeared to injure a hamstring clearing the ball. Tony Capaldi came off the bench to replace the left back.
Exeter rearranged at the interval and came out with a back four, but they were rocked by two goals in three minutes.
They had a hand in United's second goal, as Pat Baldwin tired to head back to goalkeeper Artur Krysiak. It fell straight to Constable, who coolly found the net for the second time.
United's 314 away fans were still celebrating when the next goal arrived.
Rigg did well to tee up Leven, who drilled a low shot from 20 yards into the corner past a flat-footed Krysiak.
City boss Paul Tisdale responded with a triple substitution just before the hour mark.
The reshuffled home side were gifted a lifeline midway through the half.
Damian Batt was harshly adjudged to have handled at a corner and Cureton dispatched the penalty.
Josh Parker came on for his debut shortly afterwards and with his first involvement he got in behind the defence and crossed just out of Constable's reach.
United did well to take the sting out of the game, preventing City from building up a head of steam.
They ran the clock down well, with Clarke untested, to claim a hard-earned three points.