From the next corner, a Stephen McManus effort was turned onto the woodwork by Mallo.
Mallo saved a free-kick effort from Dougie Imrie before Finnigan saw a tame effort saved by Esson.
Scott Brown curled a powerful drive wide from 20 yards and McDonald headed straight at Mallo from close-range.
From the resultant clearance Garry Brady tried another speculative effort which had Mallo scrambling to turn the ball wide.
Hughes is on the lookout for a challenge to Yves Ma-Kalamby and Mallo is out of contract with the Bairns.
Mallo has been keeping a low profile after suggesting the HUA's offer of World Cup trips had a political motivation.
However, Celtic settled and Georgios Samaras had a low drive from 12 yards turned round the post by goalkeeper Dani Mallo.
Former Celtic full-back Mo Camara, searching for a goal on his debut, rifled in a powerful drive which Mallo struggled to punch clear.
Mallo almost presented the Highlanders with a chance to level when he spilled Richie Foran's shot into the path of Rooney, but Tam Scobbie did enough to avert the danger.
But it was the home side who took the lead when midfielder Dorman was fed a through ball by Thomson and fired the ball through the legs of Falkirk keeper Dani Mallo.
Mallo was signed by Hughes for the Bairns midway through last season and the new Hibs boss has already been reunited with two former Falkirk players in midfielders Patrick Cregg and Kevin McBride.
The points were virtually secured when Mallo failed to gather Nakamura's low cross at the front post and Vennegoor of Hesselink beat substitute defender Dean Holden to the ball to bundle it over the line from four yards.
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