Play was halted when a torrential hailstorm and high winds hit the Twickenham Stoop and the crowd was warned to move clear of scaffolding at one end of the ground.
With high winds, rain, hail and lightning lashing The Stoop where advertising hoardings and assorted debris were blowing on to the pitch, referee Morris ordered the players to the dressing room.
It was announced that the game had been stopped early on "health and safety grounds".
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Even though their own hopes of reaching the last four had been virtually extinguished before kick-off, Quins had shown their pride with a disciplined display which leaves Tigers' last-four ambitions in the balance.
Former England winger Paul Sackey, flanker Maurie Fa'asavalu and number eight Tom Guest scored the Harlequins tries with fly-half Ben Botica kicking a conversion and a penalty.
Leicester replied with two penalties from fly-half Ryan Lamb, back in the side after recovering from a broken finger.
Powerful ex-England number eight Thomas Waldrom, who is bound for Exeter at the end of the season, was prominent in the early Leicester attacks as he drove holes through the Quins defence.
Lamb kicked the visitors into the lead with a fifth-minute penalty but Harlequins hit back with Sackey scoring their first try seven minutes later.
There did not look too much on when Botica passed 40 metres from the Tigers line but Sackey evaded the cover and then rounded full-back Scott Hamilton before crossing.
Botica's conversion attempt rebounded from the post but the fly-half made amends with a superb penalty into the right-hand corner to set up Quins' second try.
Winning the resulting lineout, the home pack drove Fa'asavalu over the Leicester line but the usually reliable Botica missed a second easy conversion.
Lamb's second penalty, after Harlequins were caught offside in front of their own posts, reduced the deficit to four points seven minutes before half-time but Botica cancelled it out by making it third time lucky with his first penalty five minutes later.
Leicester's hopes of overhauling Quins suffered a serious setback when Lamb's telegraphed pass was intercepted on the halfway line by Guest who left his chasers trailing on a 50-metre run to touch down between the posts, leaving Botica a simple conversion.
The final blow had left Tigers looking a well-beaten side before the premature end.
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