Hodgson grabbed a first-half try, added the conversion and landed seven penalties from eight attempts in this Premiership encounter at the Kassam Stadium.
Despite soaking up Saracens pressure for most of the match, Welsh had Sarries hanging on at the end after producing two breakaway tries from Canada wing Phil MacKenzie and full-back Tom Arscott. Henson's boot contributed their other 13 points as they collected a losing bonus point.
In the day's other match Wasps enjoyed a strange but welcome victory over Worcester as they won 10-6 at Adams Park.
The home side looked set to run away with the match after Simon McIntyre's try after just four minutes, with Stephen Jones converting and then adding a penalty 11 minutes later to put Wasps 10-0 up.
But the rest of the half was scoreless, and Worcester seemed to have got back into it just after the break though a penalty and a drop goal by Andy Goode.
Those goals proved to be the last scores of the match, however, and Wasps held on for just their second victory of the season.