Three of the tries came during a devastating five-minute spell in the first half when the Warriors defence was blown apart.
Quins fly-half Chris Malone added two penalties and two conversions, leaving Worcester heading into a six-week Premiership break just four points above bottom club Bristol.
Jones did his utmost to keep Warriors in the contest, contributing 18 points including his well-worked try.
Wing Charlie Fellows made his first start for Worcester, replacing fellow Warriors academy product Chris Pennell, while Chris Robshaw took over from injured England international Nick Easter in Quins' back-row.
Worcester coach Clive Griffiths:
"We are nearly there, but it is not happening for us at the moment.
"We lost the ball four times in broken-field situations, but we salvaged something out of the wreckage and we 'won' the second half.
"There was no panic at half-time. We were 14 points down, not 40, and in the modern game you can score very quickly. I was delighted with some of our back play.
"We want to stick to our principles of how to play the game, but sometimes the rules nullify how you want to play."
Harlequins boss Dean Richards: "David Strettle has hurt his foot. He came off as a precautionary measure, and we will have to wait and see what the results of the medical checks are.
"To come here and take five points is a big 'ask'. I don't think there are many sides who have done that.
"I am a little bit disappointed we didn't capitalise on our first-half performance, but we got the result we wanted."
Worcester: Latham, Fellows, Rasmussen, Tuitupou, Benjamin, M. Jones, R. Powell, Mullan, Lutui, Tuamoepeau, Rawlinson, Gillies, Horstmann, Wood, Talei. Replacements: Grove for Rasmussen (66), M. Powell for R. Powell (61), Fortey for Lutui (71), Horsman for Tuamoepeau (52), Cox for Horstmann (38).
Not Used: Kitchener, Crichton.
Harlequins: Brown, Strettle, Tiesi, Turner-Hall, Monye, Malone, Care, Jones, Botha, Ross, Kohn, J. Evans, Robshaw, Skinner, Guest. Replacements: Amesbury for Strettle (38),Gomarsall for Care (71), Fuga for Botha (63).
Bookmark with:
What are these?