After a first half of few chances, Josh Gowling was adjudged to have brought down Simeon Akinola in the box and Iron captain Davis slotted in from the spot.
Mariners forward Padraig Amond was then denied from close range by both keeper Tom King and defender Matt Fry.
The Iron held on to take a lead into Sunday's second leg at Cressing Road.
Part-timers Braintree, who finished third in the table and have never been in the Football League before, withstood pressure from their hosts in the second half.
Danny Cowley's side survived a penalty shout against them when Amond collided with Mark Phillips in the area.
That incident came soon after Davis had sent James McKeown the wrong way from the spot at the other end, referee John Brooks having ruled that Akinola's fine run had been halted illegally just inside the area.
The closest Grimsby came to an equaliser was through Amond, whose turn and shot from a corner was kept out by a fine reflex save by King.
Fry then deflected an effort over the bar when Amond looked certain to score for the Mariners, who have been outside of the Football League since 2010.
But Braintree, managed by a PE teacher, stifled many of the home attacks to defeat Grimsby by the same scoreline as in April.
Kenny Davis' second-half penalty gave Braintree a slender advantage from their National League play-off semi-final first leg at Grimsby.