Eliasson popped up at the death at Griffin Park to secure City their first win in six Championship matches.
It was also the Robins first success in west London since 2005 and stretched the hosts’ winless run to seven games.
Brentford’s cause was not helped when Chris Mepham saw red for a second booking just before the hour mark, with the game evenly balanced.
It started brightly for the Bees under Frank, who replaced Dean Smith, and Sergi Canos should have given them the lead after 10 minutes but bundled Henrik Dalsgaard’s low cross from the right just wide.
Top-scorer Neal Maupay was next to signal his intent with a low drilled effort that flashed inches wide of the upright.
City lost centre-back Nathan Baker to injury with 20 minutes gone after falling awkwardly at a Brentford corner.
Teenage goalkeeper Max O’Leary was City’s saviour in the first half as Brentford dominated possession but failed to find the killer instinct needed to get their noses in front.
First the debutant denied Maupay’s drive low to his right and then pulled off a reflex stop to fist an Ollie Watkins piledriver to safety after good work down the right from Romaine Sawyers.
But as the half wore on Lee Johnson’s side grew in confidence and, buoyed by the flat Griffin Park crowd, started to carve chances of their own.
Almost on half-time Brentford keeper Daniel Bentley was forced into a reflex stop of his own to deny Andreas Weimann as he tried to steer Jack Hunt’s driven cross into the far corner.
A minute later Matty Taylor’s headed goal was ruled out for offside as he strayed inches beyond Bees left-back Yoann Barbet.
But it was Barbet who had the final say of a largely tepid first half when he forced a super full-length save from O’Leary with a dipping 25-yard free-kick.
After the break City grew in confidence and saw a string of efforts fly high and wide from range before Watkins should have given the Londoners the lead after 70 minutes.
The striker smashed an angled drive back off the far post in one of the home side’s few clear-cut chances throughout the 90 minutes.
Bentley had to be at his best in the hosts’ goal to tip Jamie Paterson’s instinctive effort wide and went one better minutes later to go full length to turn Liam Walsh’s blistering free-kick to safety.
Maupay forced a great save from O’Leary with 10 minutes remaining as Brentford pressed for a winner their lack of cutting edge and discipline barely deserved.
The decisive moment came when substitute Jay Dasilva whipped in a cross to the far post and the onrushing Eliasson smashed past the exposed Bentley.