Belgium had been placed among the favourites heading into the finals, but put in a disappointing display in their Group E opener, going down 2-0 to Italy.
Saturday's performance in Bordeaux was much more akin to what is expected of a side boasting such fine individual talent and a workmanlike Ireland side struggle to cope.
Marc Wilmots' men dominated throughout, but it took until the third minute of the second half to break the deadlock when Lukaku – heavily criticised for his lacklustre display against the Italians – finished with aplomb following an assist from the excellent Kevin De Bruyne.
The second goal arrived just after the hour as right-back Thomas Meunier – in as one of three changes – delivered a perfect cross for Witsel to plant a free header beyond Darren Randolph.
And Randolph was then left further exposed as Belgium broke upfield where Eden Hazard squared selflessly for Lukaku, who calmly picked his spot to double his tally.
Victory moves Belgium off the bottom of the group and up to second, leaving them needing only a point against Sweden to qualify, while Ireland will likely require a victory against group winners Italy to have a chance of reaching the last 16.
Belgium looked invigorated by the alterations to their starting XI and it was another of those, Yannick Ferreira Carrasco, whose low cross was gathered by Randolph with Lukaku waiting to tap home inside five minutes.
Two De Bruyne set-pieces caused problems inside the Ireland box – Toby Alderweireld glancing a corner wide before a dangerous free-kick flashed across the face of goal.
And another De Bruyne delivery in the 21st minute was cleared only as far as Hazard, who blazed wastefully over from an inviting position inside the box.
Belgium continued to look the more likely to take the lead as the interval approached and Alderweireld saw a header cleared off the line by Wes Hoolahan.
And following the restart, it did not take long for Wilmots' side to forge ahead.
The ever-dangerous De Bruyne led a rapid counter down the right and fed Lukaku, who took a touch before curling a superb finish into the bottom-left corner from the edge of the area.
Belgium doubled their advantage 13 minutes later as Meunier picked out Witsel, who was left all alone by James McCarthy to rise high and nod powerfully into the goal.
As Ireland poured forward searching for a goal to get themselves back in it, Belgium once again displayed their counter-attacking prowess in the 70th minute as Hazard skipped past Ciaran Clark and teed up Lukaku for a simple finish to cap off a resounding win.