Gough (50) edged a Zaheer Khan delivery but Davies could not pouched the catch and Matt Mason took the last wicket as Essex were bowled out for 283.
Zaheer finished with figures of 9-138, the best of his career and Worcestershire skipper Vikram Solanki enforced the follow-on.
Richard Johnson was the last player to take all 10 wickets in an innings of a Championship match for Middlesex against Derbyshire in 1994.
Earlier, after Worcestershire declared their first essay on a huge 650 for seven wickets, Zaheer's devastating bowling had Essex running for covers and they were 186 for nine at stumps on day two.
Essex opener of Indian origin Varun Chopra (53) and Mark Pettini (48) shared 117 for Essex but Zaheer soon hit back with vengeance.