Nick Evans scored all of Quins' first 10 points while Tom Williams and Danny Care went over to give the visitors a 20-16 half-time advantage as Chris Pennell's try and three Andy Goode penalties kept Worcester in touch.
However, Quins pulled away after the break, Care and Evans both helping themselves to their second tries while Mike Brown also crossed. Goode added another penalty in the second period and Alex Grove went over for Worcester's second try.
Northampton also sealed their place in the semi-finals with a comprehensive 47-7 bonus-point win over Sale at a sun-drenched Franklin's Gardens.
The seven-try rout saw Jamie Elliott and James Wilson score a brace of tries each with Luther Burrell, Lee Dickson and Tom Wood also going over.
Sale, whose solitary try came from James Gaskell in the last minute, made seven changes to the side that beat Gloucester last week seemingly aiming to get as many English qualified players into the side to bump up their bonus from the RFU and Northampton took full advantage.
Bath reserved their best performance of the season for the final home game, shocking Leicester with a 27-26 win to earn an improbable bonus-point victory.
It kept Bath's hopes of a Heineken Cup berth alive - just - and left the Tigers still needing to secure a home semi-final in the Aviva Premiership play-offs.
The match-winner, with a 73rd-minute try, was captain Francois Louw, who appeared on the shoulder of replacement prop Anthony Perenise to sprint 30 metres to the corner. The hosts' tries came from Simon Taylor, Semesa Rokoduguni and Horacio Agulla, while Tom Croft and Ben Youngs crossed for Leicester and Toby Flood kicked 16 points.
Prop Halani Aulika's try-scoring antics continued as his double helped London Irish to a comprehensive 47-28 victory over London Welsh at the Madejski Stadium.
The giant Tongan tighthead now has 11 tries in all competitions and four in his last four Aviva Premiership games. His two scores, in addition to tries from Guy Armitage (two), Sailosi Tagicakibau, Chris Hala'ufia and Matt Garvey saw Irish run riot.
For Welsh, who are already guaranteed to finish bottom of the table, it was a case of what might have been as the boot of Gordon Ross and tries from the impressive Seb Stegmann and Nick Scott kept them close for a while.
England number eight Ben Morgan scored a superb solo try as Gloucester ended Saracens' six-game unbeaten Aviva Premiership run with a 28-23 win at Kingsholm.
Morgan, who recently returned to action following an ankle ligament injury suffered during England's opening Six Nations game against Scotland this season, struck midway through the second-half.
It followed two first-half touchdowns from wing Jonny May, while fly-half Freddie Burns kicked 13 points. Saracens, meanwhile, remain top of the Premiership, their losing bonus point keeping them four clear after centre Joel Tomkins scored two tries, while Owen Farrell converted both scores and landed three penalties.