PSG loanee Weah – son of former World Player of the Year George – was introduced with 20 minutes left and put the seal on the William Hill Scottish Cup holders' fourth-round victory with a finish his father would have been proud of.
Scott Sinclair missed a first-half penalty but made amends with a two smart finishes as Celtic eased past Ian Murray's League One minnows to book their place in Sunday's last-16 draw.
While Weah started on the bench, Brendan Rodgers threw fellow January signing Oliver Burke straight into his line-up as Odsonne Edouard again sat out through injury.
Yet the contribution of the on-loan West Brom winger during the opening half hour was minimal as Airdrie did their best to make life difficult for the hosts.
The Diamonds worked tirelessly and did well to get their blocks in anytime Celtic threatened to cut loose early on – but with the hosts remaining patient it was only a matter of time they opened them up.
Celtic's relentless probing eventually led to a penalty after 32 minutes.
Anthony Ralston had a shot blocked after tip-toeing his way past three opponents into box and then laid off for Emilio Izaguirre, who was tripped by Chris O'Neil.
Airdrie skipper Sean Crighton was not happy with how easy the Honduran went down and referee Andrew Dallas was forced to intervene to prevent a melee getting out of hand.
The delay seemed to disturb Sinclair and his spot-kick was just far enough inside David Hutton's right-hand post to allow the Airdrie number one to pull off a brilliant stop.
The winger's day looked to be going from bad to worse as he somehow smashed a Ralston cross over the crossbar from point-blank range moments later.
But the Englishman made it third time lucky in the 36th minute as he did find the net.
James Forrest's cut-back teed-up Burke and while his shot was not the cleanest it still forced Hutton down to his right. The keeper could only palm the ball back into traffic and Sinclair pounced for the opener.
Yet Celtic were grateful to their own keeper Scott Bain not to be pegged back just before the break.
Scott Stewart forced his way past Izaguirre and threw across a great ball for Kyle Wilkie, who was denied as Craig Gordon's understudy pulled off a heroic block before leaping up to keep out two more efforts amid a chaotic scramble.
But progress to the next round was secured 11 minutes into the second half as Sinclair grabbed another close-range strike.
Dedryck Boyata got his head to a Ryan Christie corner, Hutton's reflexes prevented a ricochet off team-mate Kieran Millar flying into the Airdrie net but that left him powerless again to prevent Sinclair scoring.
Sinclair had the ball in the net again after 68 minutes as he jumped on another rebound but he was denied his hat-trick by the linesman's offside flag.
Sinclair made way for Weah midway through the second period and the 18-year-old took over the scoring duties with seven minutes left as he raced onto Boyata's through ball before calmly steering home the third.