It was a third successive victory for Steve Cotterill's side and came despite the Robins being reduced to ten men early in the game at London Road after defender Adam El-Abd was sent off following an off-the-ball incident.
Elliott played the full match for the Sky Bet League One side.
The win also saw Robins move up to 15th place, five points ahead of the drop zone.
Elsewhere in League One, midfielder Hayden Mullins saw his Notts County side slip to a 3-1 defeat at home to MK Dons.
Mullins completed the full match for the Magpies.
The Blues duo of Nick Townsend and Charlee Adams were in action for Lincoln City.
The Skrill Premier League side suffered a first defeat in six matches as they were beaten 2-1 at home by Welling. Townsend and Adams played the full 90 minutes.
There was more success elsewhere for Amari'i Bell.
The young Blues left-back made his loan debut for Skrill Premier side Kidderminster Harriers against Macclesfield.
Harriers recorded at 2-1 at Aggborough, with Bell entering the proceedings in stoppage time at the end of the game.
Finally, in the Skrill North division, full-back Ryan Higgins played the full game for Nuneaton in their 1-0 home defeat to Vauxhall Motors.