The 30-year-old was trapped lbw by former England seamer Ryan Sidebottom in the third over of the four-day game.
James Hildreth made 89 in the MCC's 221, with Yorkshire closing on 82-2.
It is Cook's first match since the one-day series in Sri Lanka, after which he was dropped and replaced as ODI captain, missing the World Cup.
The Essex batsmen has claimed England's poor showing at the tournament - which they exited at the pool stage having won just two of their six matches - proved the selectors were wrong to replace him with Eoin Morgan and omit him from the squad.
Cook has been named as captain for England's three-Test tour of the West Indies, which gets under way on 6 April.
He began his season, which also sees the visit of Australia in the summer, by playing for the MCC in a Twenty20 match against Sussex on Friday, top scoring with a 26-ball 35 in a 19-run defeat.
However, in the traditional County Championship season curtain raiser, his wicket left the MCC wobbling on 7-2 after skipper Nick Compton had edged the first ball of the match from Sidebottom behind to Jonny Bairstow.
Cook's former England opening partner Michael Carberry steadied the innings with a composed 36 before James Hildreth (89) and Worcestershire captain Daryl Mitchell (54) shared a stand of 79 as they threatened to take the game away from Yorkshire.
But Hildreth's 118-ball knock was ended by Tim Bresnan, who trapped him lbw, and wickets began to tumble as only Zafar Ansari showed any resistance with 24 not out as MCC collapsed from 160-3 to 221 all out.
Yorkshire then reached 82-2 at stumps with Adam Lyth unbeaten on 53.
England Test captain Alastair Cook was dismissed for just three on his return to first-class cricket playing for the MCC against Yorkshire in Abu Dhabi.