Scottish Premiership Review: Hamilton go top with historic victory over Celtic
Alex Neil's side had to withstand a barrage of pressure from the hosts throughout the match, but it was Crawford who netted the only goal of the game early in the second half.
John Guidetti, Anthony Stokes and Jason Denayer were all denied by Michael McGovern in an opening half-hour dominated by the hosts, with the Hamilton goalkeeper's one-handed stop from Denayer particularly impressive.
Early chances again fell to Guidetti and Stokes after the break and their profligacy was punished on 48 minutes as Crawford struck his fifth of the season.
The forward collected a cross from Steven Hendrie and fired past Craig Gordon from 14 yards to open the scoring.
Celtic pushed for an equaliser but Kris Commons, Leigh Griffiths and Stokes proved unable to convert chances as Hamilton held on to beat Celtic for the first time since 1989.
The defeat leaves Ronny Deila's side sixth in the table, six points behind Hamilton with one game in hand.
Inverness Caledonian Thistle, meanwhile, had to come from a goal behind to move into third place with a 1-1 draw against Ross County.
Yoann Arquin's first goal of the season gave County the lead on 21 minutes.
However, Marley Watkins equalised four minutes into the second half, converting a pass from Ryan Christie.