The shocked 21-year-old rookie was unhurt in the incident between turns two and three but the chassis had to be replaced, sidelining the Briton until Saturday's final session before qualifying.
Some estimates put the likely cost to Williams of the damage at more than half a million dollars.
"There clearly should be some recompense," a disappointed deputy team principal Claire Williams told Sky Sports television.
"I know that there's precedent, Haas I believe were compensated after it happened to them in Sepang (Malaysia) a few years ago so clearly we are going to be talking to them (the circuit) but I can't put a number on it at the moment," she added.
The former world champions are already facing plenty of problems with an uncompetitive car that is some way off their rivals' pace and that was late to testing.
They are last in the standings and neither Russell nor Polish team mate Robert Kubica have scored a point in three races this year.
Russell won at the Baku street circuit last year on his way to the Formula Two title but Kubica has never raced there before and Williams need all the track time they can get.
"I just hit it, the biggest shock went through my body. The whole engine just turned off, it's ruined the floor," said Russell. "It was just on the normal racing line and has ruined my session."
The session was red flagged and then abandoned just 12 minutes into the scheduled 90 as organisers and track officials worked to check some 320 manhole covers and carry out repairs.
"They're going to fix it (the track) now but of course it shouldn't have happened," said Red Bull's Max Verstappen. "Running over a cover like that at 300 kph is not nice."
Race director Michael Masi said a preliminary investigation indicated one of the clamps beneath the manhole had failed.
In more farcical scenes, the mobile crane truck carrying Russell's stricken Williams back to the pits then hit the underside of a sponsor hoarding spanning the track, leaking hydraulic fluid onto the car.
Ferrari's two drivers were the only ones to set a time, with Charles Leclerc ahead of Sebastian Vettel. Replays showed the Monegasque running over the cover, potentially dislodging it, just before Russell.
Five-times world champion and overall leader Lewis Hamilton, winner in Baku last year for Mercedes, did not seem to mind the lack of action, however.
"How could they have not checked and sealed the drains?" the Briton asked on his Instagram feed.
"Oh well, (it) just gives me extra time to catch up on Game of Thrones," added the 34-year-old, who has been binge-watching the popular fantasy series.
The second practice session started as scheduled. - Reuters