The champions had to play second fiddle to Falkirk for most of the afternoon but had the cutting edge that the home side lacked.
McDonald's headed goal - his 19th of the season - brought the three points to cut the gap at the top to four points ahead of Wednesday's rescheduled match against Motherwell.
Robert Olejnik started in goal for Falkirk with Tim Krul injured but he had little to do in the 90 minutes as Celtic struggled in the blustery conditions.
Falkirk looked the more dangerous and forced a couple of early free-kicks, but their deliveries were poor and Artur Boruc was untroubled.
Arnau Riera tried his luck from distance but it was straight at the goalkeeper and Boruc gathered easily, before Graham Barrett saw a shot blocked for a corner by Lee Naylor.
Pedro Moutinho was causing Celtic some concern and he set up a chance for Kevin McBride but his effort was too high.
Boruc produced an excellent save low to his right to touch a Darren Barr effort past his post after a curling free-kick from Tom Scobbie.
But chances were few and far between in a first half that became bogged down in the midfield with Celtic's creative players kept subdued.
Barrett had a great chance to make the breakthrough for Falkirk when he beat the Celtic offside trap after great work from Dean Holden, but Boruc was off his line quickly to make a fine save.
Holden's efforts to create the chance came at a cost as he was stretchered off with a suspected broken leg after the tackle from Stephen McManus in the build-up.
Celtic took the lead against the run of play in first-half injury time.
McDonald fed Shunsuke Nakamura before darting into the box and meeting his team-mate's cross to nudge a header past Olejnik.
It was tough on Falkirk after all they had put into the first period and they opened the second half on top as they went in search of the equaliser.
Riera and Barrett both had good shooting chances early in the second half but dragged their efforts wide.
Falkirk dominated for long stages of the second half but failed to put consistent pressure on Boruc and Celtic defended well.
Celtic improved after the introduction of Evander Sno midway through the second half as Falkirk seemed to run out of ideas.
The home side had a strong appeal for a penalty in the 73rd minute when McBride looked to be caught by Massimo Donati but referee John Underhill was not interested.
Jan Vennegoor of Hesselink almost made it 2-0 in the final minute but his shot struck the post with Olejnik beaten.