Taylor Miles' free-kick gave Braintree an early lead, only for Ben Clark to level from close range soon after.
Miles' 20-yard shot put the Iron back in front, before Heed centre-back James Curtis was sent off for bringing down Michael Cheek before the break.
Mitch Brundle's shot made it 3-1 and the Iron held on for the win, despite Alex Whitmore's goal for the hosts.
Braintree are now one point clear of sixth-placed Eastleigh, while Gateshead are 10th in the table.
Braintree Town moved into the National League play-off places with a hard-fought victory at 10-man Gateshead.