Solanki's knock came off just 59 balls and included 12 fours and two sixes, as he, Davies and Ali (who hit three sixes) feasted on Hampshire's inexperienced attack to leave the visitors anchored to the bottom of the South West table.
Hampshire failed to build valuable partnerships and Jones eventually rattled through the tail, taking the last four wickets cheaply and easily passing his previous one-day best of 3-19 for Glamorgan against Lancashire in 2005.
Worcestershire paceman Simon Jones told BBC Hereford and Worcester:
"We felt we have been capable of doing something like that since the start of the season.
"With a couple of the guys back in, and the weather being a lot better, we feel we can make progress in this competition.
"Today was proof of the quality we have, and how we can play as a team."
Hampshire manager Paul Terry told BBC Radio Solent:
"Our skill level was really disappointing. We've got a lot of work to do.
"Lamb took a great opportunity batting up the order and scored a really good 84. That was pretty much the only positive.
"I don't think we made them work at all with our bowling. They were able to have a pretty easy ride."