The Brazilian struck twice to ensure leaders Marseille remain within their sights at the top of Ligue 1 and continue the club's unbeaten start to the campaign
Third-placed Bordeaux produced a resolute display for much of the first half at Parc des Princes on Saturday, but PSG ultimately came out on top in a match which saw both sides reduced to 10 men.
The first to be dismissed was PSG defender Gregory van der Wiel in the 28th minute, with the Netherlands international earning his first red card for the club somewhat harshly for a late challenge on Wahbi Khazri.
Bordeaux were then reduced to 10 men nine minutes later, as Andre Biyogo Poko was sent off for a reckless tackle on Marco Verratti and that seemed to spark the hosts into life.
Lucas got the breakthrough with his first spot-kick just before the break after good work by Verratti and the Brazilian doubled his tally from 12 yards early in the second half.
Second-half substitute Ezequiel Lavezzi added a third late on, as PSG continued their unbeaten start to the season and moved four points behind pacesetters Marseille, who face the tough task of heading to Lyon on Sunday.
PSG coach Laurent Blanc entrusted 21-year-old Jean-Christophe Bahebeck with filling the void left by the injured Zlatan Ibrahimovic and suspended Edinson Cavani in attack.
And although the forward looked lively, PSG failed to penetrate the Bordeaux defence during the early exchanges, leaving Bahebeck to feed off scraps.
After a low key opening, the match began to liven up and Bordeaux defender Nicolas Pallois headed a glorious chance well wide following a fine free-kick delivery from Jaroslav Plasil after 20 minutes.
PSG looked to be up against it eight minutes later, when Van der Wiel was shown a straight red card for a late tackle on Khazri.
The former Ajax man was baffled by the decision and Blanc responded by replacing Bahebeck with right-back Serge Aurier.
Bordeaux's numerical advantage lasted less than 10 minutes, though, as Biyogo Poko followed Van der Wiel down the tunnel for a wild challenge on Verratti and David Luiz's effort from the resulting free-kick crashed against the post.
Biyogo Poko's sending off clearly reinvigorated PSG and the champions took a lead into the break as Lucas coolly slotted in a stoppage-time penalty after Verratti was tripped by Cedric Yambere.
It took the home side just four minutes to double their lead in the second half, with Lucas once again holding his nerve from 12 yards to stroke home after being hauled down by Pallois.
Bordeaux produced a response as they attempted an unlikely comeback, but PSG stood firm with Luiz making a vital block to deny Diego Rolan, before Salvatore Sirigu tipped an audacious Emiliano Sala effort just over.
Any hopes of a Bordeaux fight-back were firmly extinguished nine minutes from the end as Lavezzi, Lucas' replacement, flicked in Blaise Matuidi's left-wing cross, wrapping up a comfortable win and throwing down the gauntlet to Marseille.