A double from Zak Rudden inside the first 15 minutes - the opener coming from the penalty spot - put daylight between the sides before Finlay Robertson struck late in the second half to cap off the victory.
Dundee have now won six matches in a row across all competitions but the only frustration for Rudden was that he didn't leave the Falkirk Stadium with the match ball.
"I'm delighted and we're delighted for the fans as well," the 22-year-old striker told BBC Alba. "It's good to get two goals and, overall, a good performance from the boys.
"I should have got a couple more. I should have got that hat-trick - that was a poor one from me."
The game was five minutes old when Falkirk left-back Leon McCann bundled Paul McMullan to the ground in the box and referee Calum Scott pointed to the spot.
Goalkeeper PJ Morrison got a big hand to Rudden's spot-kick but was unable to prevent the effort from nestling in the corner of the net.
Ten minutes later it was 2-0 to the Championship side as a delightful low cross from Luke McGowan was met by Rudden who made no mistake from six yards out.
Falkirk boss John McGlynn, who won the competition with Raith last season, made a triple substitution at the break as the League One side looked for a way back.
But Dundee goalkeeper Adam Legzdins was rarely troubled and Robertson curled home a stunning effort from 20 yards to finish the Bairns off.