First-half tries from Rory Best, Johnny Sexton and Jack Conan paved the way for victory, with Keith Earls grabbing the crucial fourth after the interval.
Joe Schmidt's side now needs to beat table-topping Wales in its final match in Cardiff and hope England fails to get the better of Scotland if the trophy is to remain in its hands.
Ireland, already without Robbie Henshaw and Dan Leavy, also lost Rob Kearney to injury as Jordan Lamour came in at full-back, but the early signs suggested it did little to undermine its fluency.
Best put the finishing touch on a move started by his own line-out throw in the third minute and Sexton added the extras.
France thought it had provided a swift response, but Thomas Ramos knocked on before pulling a cross-field kick out of the air to score.
Concerted Ireland pressure did not get its reward until just before the half-hour mark, Sexton deftly exchanging passes with Garry Ringrose before darting through a gap to touch down for his first Test try since 2015.
Having lost to England in its previous home game, Ireland all but made sure of avoiding the same fate this time around when Conan bulldozed his way over the line just before the break.
It took a while for Schmidt's charges to get going in the second half, but Earls sprinted away for the bonus-point score to set up a tantalising final round of the 2019 Six Nations.
France only stirred to life in the dying embers of the contest, with Yoann Huget's late try – which came while Dorian Aldegheri was in the sin bin after repeated infringments – followed up by another from Camille Chat beyond the 80th minute.