Akinfenwa and Weston struck either side of the break to extend the Gills' unbeaten run to nine games and give Andy Hessenthaler's men a desperately needed three points in their promotion push.
Bradford squandered two early chances through Luke Oliver's header and Gareth Evans' shot, neither of which was on target.
Kevin Ellison also brought a fabulous save out of goalkeeper Alan Julian but instead it was Gillingham who took the lead two minutes before the break as Weston crossed for Akinfenwa to head home his ninth of the season.
Gillingham doubled their lead just seconds after half-time when Weston fired home from 25 yards.
Julian was again called on to preserve Gillingham's advantage and keep out Bradford substitute Tom Adeyemi's long-range effort while James Hanson missed the target for the visitors with a close-range header.
Source: PA
Source: PA