Grace, who made a century in his last game at the Home of Cricket, also for MCC against Minor Counties in 2012, struck 103 from 134 balls as the home side posted 275 batting first.
Former Somerset wicket-keeper Steve Snell scored 76 in a fourth-wicket partnership of 130 with Grace, while Dravid made 14 batting at number four. David Adnams was the pick of the bowlers for the tourists, collecting 3/43.
The visitors were always behind the run of play in response, despite David Jacquier making 75 at the top of the order.
Will Jones gave MCC a brief scare in the latter stages when he hit eight fours in a 37-ball knock of 54, but he was dismissed by captain John Stephenson as the hosts finished strongly.
Stephenson finished with 4/53 from his 10 overs, while Robert Franklin was the pick of the bowlers with 4/23. Rupert Kitzinger, an MCC tourist in Uganda last year, also picked up two wickets.