Falkirk survived lengthy spells of pressure from their hosts after playing the whole second half at Airdrieonians with 10 men.
Paul Sheerin’s side had scored two quick goals just after the midway point of the first half and looked relatively comfortable, with the Diamonds perhaps rueing earlier missed chances.
But skipper Hetherington walked just a minute or so from the break after a second yellow card, which spelled potential trouble for the Bairns.
However, a regrouping to shore things up in the middle of the park and compact defending saw them safely through to victory.
Both sides showed some skilful touches in a low-key opening quarter of an hour, with some neat midfield passing and calmness in defence by both sides.
The Bairns had a chance as Aidan Keena found Callumn Morrison with a great pass but the winger's effort was blocked.
Craig McGuffie created a dangerous opening in the 16th minute when he got past easily Scott Walker on the left but his cross was over-run by Hetherington and drifted out of play.
Hetherington was cautioned on 22 minutes after a midfield tangle with an opponent and Dylan Easton showed Airdrieonians' intent with a low shot in 25 minutes which was deflected past for a corner – one of 10 awarded to the hosts during the game.
A good chance presented itself from that corner, with an overhead kick from Callum Smith and a header just over the bar. But then Falkirk, who had been putting together a string of interesting-looking breaks, hit their hosts with two goals in as many minutes.
A 27th-minute shot from Aidan Nesbitt was saved by ‘keeper Max Currie but only pushed wide into the path of Callumn Morrison, who delivered a simple finish.
Another good Falkirk move saw Aidan Keena make it 2-0 on 29 minutes with a beautiful chip over Currie.
Dylan Easton was presented with a shot after 35 minutes when a ball across was missed by the Falkirk defence and ‘keeper Robbie Mutch, but the Bairns got enough bodies back to clear the danger.
Airdrie knew they could ill afford to concede again before the interval and excellent work on the right, after 38 minutes, between Calum Gallagher and Gabby McGill, saw a cross sent in. But Callum Smith just failed to connect with his header.
The Bairns were pared back to 10 men just before half time when Hetherington collected a second booking for a bad tackle. Airdrie needed help from somewhere and it was just possible this could be it.
Falkirk still had time to catch Airdrie on the break, however, and Nesbitt brought out a save from Currie with a fine low right-foot shot.
The pressure anticipated by the Diamonds at the start of the second half was realised after only a couple of minutes as Adam Frizzell produced a marvellous save from Currie.
McGill then brought Airdrieonians back into it on 50 minutes, showing good composure to score from close range.
The first change was made on 56 minutes by Falkirk, with McGuffie making way for Ernaldo Krasniqi.
Falkirk collected another yellow card soon afterwards for a dubious challenge by Charlie Telfer.
Rico Quitongo made a fine pass to Dylan Easton, whose 65th-minute cross from the left for Airdrie was headed towards the target by Gallagher but Mutch held it.
Airdrie continued to pressurise as Falkirk replaced Keena on 68 minutes with Samuel Ompreon. Scott Agnew came on for Airdrieonians after 71 minutes for Gabby McGill, while they also replaced Gallagher five minutes later with Jordan Allan.
A couple of home players got in each other's way and failed to capitalise on a great cross by Scott Agnew, while at the opposite end, Falkirk’s Morrison weaved past a couple of players in the opposing box before shooting wide, with 10 minutes to go.
Airdrie's Craig Watson was cautioned for fouling Morrison, after which Scott Walker was replaced in 82 minutes by Hearts loanee Scott McGill. Jaime Wilson took over from Morrison after 86 minutes for Falkirk.
Airdrie continued to shovel on the pressure but Falkirk defended robustly. A final chance for the visitors to square it was thrown away five minutes into added time when Smith headed McGill’s cross on to the roof of the net.
Airdrieonians: Currie, Quitongo, Fordyce, McCabe, Gallagher (Allan 76), Easton, Smith, Watson, Walker (S. McGill 82), Frizzell, G. McGill (Agnew 71). Subs (not used) Cantley, Ritchie, McDonald.
Falkirk: Mutch, Hall, Miller, Morrison (Wilson 86), Hetherington, Nesbitt, McGuffie (Krasniqi 56), McCann, Keena (Ompreon 68), Telfer. Subs (not used) Martin (GK), Ross, Weeks, Williamson.