Championship winners Welsh had needed to go through a Rugby Football Union appeals process just to get this far, but any jubilation at reaching the top flight looked set to be crushed as two Thomas Waldrom tries helped Leicester to a 17-0 lead.
However, the hosts showed great courage, and no little skill, to fight back through tries from Tom Arscott and Hudson Tonga'uhia and they trailed by just seven at the break.
But further scores from Geoff Parling, Julian Salvi and Vereniki Goneva sealed Leicester's victory, with fly-half Toby Flood converting all five tries and a penalty for a 13-point haul.
Gordon Ross' penalty was all Welsh could muster in the second stanza.
Flood and Ross had both missed early penalty opportunities, but the England fly-half made no mistake in the eighth minute after Welsh had folded in at a scrum.
The first try soon followed with the home side's inability to hold on to possession already proving key.
Tigers took the play through several phases and the Exiles resistance broke with Flood's long pass putting Tom Youngs through a hole to link with Parling and Anthony Allen to send Waldrom over in the corner.
Flood added the touchline conversion, and another threatening Leicester attack was only stopped by Joe Ajuwa's high tackle on a rampaging Manu Tuilagi.
Having used the rapier for their opening score, the Tigers switched to the bludgeon for their second as two huge rolling mauls splintered the Welsh pack to smithereens, with Waldrom again the beneficiary.
Welsh bravely hit back after stripping Leicester of the ball on the edge of the Tigers 22.
Scrum-half Jonah Holmes provided the impetus for Arscott to slide between two defenders and dot down for a debut try to make the score 17-5.
Having looked in danger of an opening day hiding, the hosts fightback gathered momentum just before the break.
Moments after Holmes had gone close to the line with a break down the left, centre Tonga'uhia caught the Leicester defence sleeping at a ruck to canter over in the right-hand corner and trim the deficit to seven points.
With some choice words from Richard Cockerill doubtlessly ringing in their ears, Tigers reasserted their authority within 90 seconds of the restart.
Another powerful maul sucked in the home defence, and England lock Parling had an easy finish out wide as Geordan Murphy and Tuilagi put him over.
Flood again converted, although his charged down clearance from the restart led to Ross knocking over a penalty for Welsh.
But the Tigers' bonus point try quickly arrived as a series of pick and drives took the visitors through the heart of the Exiles defence, with flanker Salvi crashing over under the sticks for a 31-13 lead.
Goneva then rounded off matters with his first Leicester try as he escaped past two defenders despite space being at a premium down the left.