The home side opened a comfortable 19-9 lead at the break thanks to four penalties from Steenson and his conversion of the only try of the half, which came from hooker Jack Yeandle. Gavin Henson replied with three penalties, but the visitors were put to the sword in the second half.
Further tries from skipper Tommy Hayes, Damian Welch, Sireli Naqelevuki and Jack Nowell, all converted by Steenson, steered the Devon side to their deserved bonus-point victory.
Phil Mackenzie scored the try for the Exiles which was converted by Henson.
Neither side had won in their last five league outings heading into the contest, while the Exiles had not tasted victory at Exeter since a National One clash in October 2000.
The Chiefs stay eighth and London Welsh are now just three points above bottom-of-the-table Sale Sharks.
Two tries from Jamie Elliott has kept Northampton in the hunt for a top-four finish after a narrow 25-23 victory against Bath at Franklin's Gardens.
With 20 minutes to go, Northampton looked to be heading towards a routine home win at 20-9 in front, but young referee Luke Pearce sparked the game into life by awarding a controversial penalty try and when centre Semesa Rokoduguni followed up with another, Bath led 23-20 with 10 minutes to go.
But England discard Ben Foden sent Elliott over for his second try with seven minutes left and Northampton clung on for a vital win.
In the late match Saracens produced a stunning second-half fightback to beat great rivals Leicester 32-27 in snowy conditions at Welford Road.
Trailing 17-6 at half time, Saracens scored four tries in a one-sided second-half to stay in second place in the Aviva Premiership.
It was their fourth successive win at Leicester, who have suffered two consecutive defeats having lost at champions Harlequins last week.
Saracens' tries came from wingers David Strettle (two), James Short and full-back Chris Wyles, with fly-half Charlie Hodgson kicking 12 points.
Julian Salvi and Dan Bowden scored for Leicester, who were also awarded a penalty try right on full-time when Strettle deliberately knocked on and was sin-binned.