Early on, Jordan Thomas was unfortunately not able to carry on due to an injury he picked up.
A set piece from Archie Gray met the head of Luis Fernandez, but his header was straight at Jack Sims.
Just after the half an hour mark, The Irons took the lead through some nice intricate passing, with Matthew Ward's shot just looping over Desbois, who got a hand to the strike but could not keep it out.
Right before the interval, Braintree Town scored a second, as Ward's cross was headed in by Shaquille Coulthirst to double their lead.
Hampton & Richmond was inches from pulling a goal back, after a dangerous ball was whipped into the penalty area, with Sam Dryer latching onto it, but his effort hit the crossbar and bounced away from danger..
The away side managed to pull a goal back, with a wonderful volley from Joshua Anifowose, that flew into the net past a helpless Jack Sims.
Into stoppage time the game went and Ben Watt's cross was turned in by Coulthirst for his second and Braintree's third.
Despite Hampton & Richmond's late efforts, they were not able to prevent defeat in this match.
The Beavers remain second in the division, whilst Braintree Town move up to eleventh.