Baker scored one and played a big part in another as Coventry made the most of stand-in goalkeeper Seb Brown's error to advance to the second round.
Kick-off was delayed until 8pm after a traffic accident held up some of the Dons squad, with the home side forced to replace regular goalkeeper Ross Worner with deputy Brown.
And it was Brown's error that allowed Callum Wilson to equalise Michael Smith's opener before Baker ensured the League One side defeated their League Two opponents.
The Dons should have led five minutes before half-time when Peter Sweeney slipped in Smith only for the striker to fire at goalkeeper Joe Murphy's feet.
Murphy earlier denied Harry Pell and George Porter and the goalkeeper was forced to deny Smith again at the start of the second half.
However, Murphy was finally beaten after 54 minutes when Smith headed in from Porter's cross, but six minutes later the Sky Blues led.
First Brown fumbled Cyrus Christie's cross allowing Wilson to poke the ball over the line and then Baker finished a far-post cross.
And Brown's miserable night continued when Baker's free-kick deflected in off Callum Kennedy in the wall in the 70th minute.