Racing trailed 16-6 at half time after New Zealand scrum-half Byron Kelleher touched down for the hosts with David Skrela transforming and adding five penalties and a drop goal to their tally during the game.
But the visitors fought back after the break sparked by a try from Fijian winger Albert Vulivuli after 52 minutes, with Steyn kicking three penalties in the last eight minutes.
Racing now have 28 points from nine games, one point ahead of Toulon, 22-15 winners over Castres, Montpellier and Toulouse, who are all on 27.
Coach Simon Mannix hailed Steyn's efforts.
"We had a profitable second half which finished well thanks to a great performance from the boot of Francois Steyn," said Mannix.
"Hats off to him because the three he got over in the last eight minutes allowed us to get the bonus point.
"The annoying thing was that every time we scored we lost points just afterwards. It's inexcusable at this level."
Stade had made a modest start to the season in winning four and losing four of their opening matches but this time they went for the kill to move out of the lower reaches of the table and hand Galthie's men their third defeat.
Montpellier did push to make a match of it after the restart and temporarily camped in their rivals' half, with Timoki Nagusa going over and Mamuka Gorgodze then adding a second collective effort.
But the Parisians held on to remain eighth, aided by three Lionel Beuxis penalties.
Elsewhere, Clermont have designs on the top six after handing Agen a seventh defeat 45-19, sending the latter bottom after Bourgoin beat Brive 13-3.
Reigning champions Perpignan, beaten twice to date, edged La Rochelle 21-16 to move into fifth, as Biarritz regained their Basque supremacy with a 22-19 success at Bayonne thanks to scrum-half Dimitri Yachvili who scored all their points.
On Friday, Jonny Wilkinson led the way with the boot as he helped Toulon go joint top with a 22-15 win -- their fourth in a row -- over visiting Castres who sink into the bottom five.
Wilkinson took the game by the scruff of the neck at the Stade Mayol, whose Delangre stand offered up the incongruous sight of English fans in the 13,500 crowd sporting Admiral Nelson-style bicorne headgear.
Adding to the English flavour, alongside several renditions of 'God Save the Queen', was referee Andrew Small.
The home-from-home atmosphere clearly proved to Wilkinson's liking as the World Cup winner landed penalties on 10, 22, 28 and 51 minutes, interspersed with a 14th-minute drop-goal to bolster morale ahead of a European Cup tussle with Ospreys.
Source: AFP Global Edition