Rashid, unbeaten on 120 overnight, went on to make 180 before he was caught behind off Peter Trego, while Rich Pyrah added a quickfire 55 off 54 balls to propel the White Rose to 505 for nine.
Somerset's openers Marcus Trescothick and Nick Compton responded in a positive manner, with the former reaching 53 before being trapped in front by Jack Brooks.
England batsman Compton was unbeaten on 33 to help Somerset reach 92 for one.
The bowlers also toiled at Edgbaston, where Middlesex racked up 284 for three on day one against defending champions Warwickshire, largely thanks to opener Sam Robson's unbeaten century.
Robson followed up his 129 in the second innings of the draw against Surrey at Lord's last week with 136 not out for Middlesex, who won the toss, today, while Chris Rogers, Joe Denly and Dawid Malan could not prosper despite making decent starts.
Jeetan Patel (two for 53) was the pick of a Warwickshire attack missing Chris Woakes and Chris Wright - due to England Lions duty - and the injured Keith Barker.
Tim Phillips took three wickets and Alastair Cook scored an unbeaten fifty to give Essex the edge against Lancashire on day two of their LV= County Championship Division Two clash at Old Trafford.
Slow left-armer Phillips took three lower order wickets, with David Masters and Reece Topley taking two scalps apiece as Lancashire were all out for 177.
Essex then built on their 49-run first-innings lead to reach 120 for two at the close, with England captain Cook contributing eight fours in an unbeaten 57.
Jimmy Adams came to Hampshire's rescue with a gutsy 74 not out as his side were reduced to 165 for eight before rain ended the opening day at Bristol.
At one point the visitors were 97 for eight, but Danny Briggs (25 not out) joined skipper Adams in a fighting unbroken ninth-wicket stand.
Half-centuries by Matt Pardoe and Thilan Samaraweera took Worcestershire to 198 for three on a rain-shortened first day against Leicestershire at New Road.
- Sports & Recreation
- Nick Compton
- Marcus Trescothick