The Chiefs scored four of their six tries in the final quarter, with Nic Sestaret and Jack Nowell both going over twice, to blow away their hosts who had led 9-8 at half-time.
Will Carrick-Smith and Mark Foster touched down for Exeter before their late blitz, with Henry Slade adding 12 points from the boot.
The Exiles, who retained 10 players from the team that met Bath last time out, relied on Gordon Ross to kick over all of their points.
The hosts had no answer to the young Chiefs who were a revelation, showing huge enterprise, and had they showed a little more composure in the first half then Welsh would have suffered an even greater defeat.
Exeter did not include any of their players who started last week's Premiership game against Worcester and they initially found themselves in an arm wrestle.
London Welsh took the lead after six minutes when Ross kicked a 20-metre penalty but Slade soon responded with an even simpler one for the visitors.
The first quarter was evenly contested but Exeter came closest to scoring. One of their forward drives actually crossed the line but their effort was ruled out by the TMO and then following penetrative runs from Joel Conlon and Myles Dorrian, Junior Poluleuligaga was pushed into touch a metre short by some desperate home defence.
Against the run of play, Welsh regained the lead when Exeter infringed at a scrum 25 metres out and Ross made no mistake with his kick to give his side a 6-3 lead after 25 minutes.
Three minutes later, the more enterprising Chiefs deservedly scored the first try. Slade's miscued kick went straight to his wing Foster who ran strongly down the left flank deep into the Welsh 22. Exeter recycled the ball and Poluleuligaga's pass sent Carrick-Smith over for the try which Slade was unable to convert.
Welsh relied on their forward drives and the tactical kicking of Ross to have the better of the latter period of the first half and they were rewarded when just before the break, Ross kicked his third penalty to give them a 9-8 interval lead.
Early in the second half, Slade put the Chiefs back in front with a 45-metre penalty before they scored an excellent second try.
Chris Budgen got them moving forward with a 10-metre drive; the backs were then given their opportunity for Sam Hill to produce an exquisite dummy to send Foster in for an easy try which Slade converted.
Ross kicked two penalties in quick succession but Welsh still trailed 15-18 going into the final quarter.
Exeter then put their foot on the accelerator.
Their burst began with a skilfully created try scored by Nowell, who outflanked the cover defence.
With 12 minutes to go Sestaret went over from close range before the French wing, put away by the lively Poluleuligaga, raced in for his second.
Two minutes from time, Nowell collected his second and Exeter's sixth, which was converted by Slade, as the Welsh were ultimately routed.