Leinster number eight Heaslip assumed the captaincy after lock Paul O'Connell was sidelined by a chest infection on Saturday night.
Tries from wing Andrew Trimble and full-back Rob Kearney cemented victory in head coach Joe Schmidt's first Six Nations clash.
Greig Laidlaw landed one of two penalty shots to hand Scotland parity at 3-3 in the opening quarter of this Six Nations showdown.
The Edinburgh half-back cancelled out Johnny Sexton's penalty, after missing his first opportunity after just five minutes.
Laidlaw wasted the first scoring opportunity, striking an early penalty against the post.
Chris Henry was penalised for sealing off a ruck, and that after an eventful opening for the Ulster seven, first charging down Duncan Weir then winning a fine turnover.
Scotland showed their attacking intent right from the off with sustained phase play, only for Laidlaw to strike the upright with that first shot off the tee.
Stuart Hogg then cut the line to feed Weir, but again Ireland stole the ball at the breakdown.
Ireland powered close from a driven line-out, before squeezing Scotland at the five-metre scrum.
Scotland thought they had completed the clearance job after stealing a line-out, only for skipper Kelly Brown to concede a kickable penalty for not rolling away in the tackle.
Racing Metro fly-half Sexton duly slotted the game's opening points, before Henry was just unable to hold a smart offload from O'Driscoll.
Scotland pressed from the scrum, eventually winning a penalty when Devin Toner failed to roll from a tackle.
Laidlaw slotted the goal to level the scores.
Andrew Trimble's 13th international try ensured Ireland turned around with an 11-3 half-time lead in their RBS 6 Nations opener against Scotland.
The UIster wing raced into the right corner for the first try of the match after Ireland had a score for number eight Jamie Heaslip ruled out.
Johnny Sexton bisected Scotland's defence to send Heaslip down the left wing, only for Television Match Official Carlo Damasco to rule the Ireland captain had been hauled into touch.
Trimble then claimed his first Ireland try since last summer, on his 51st cap, running in after two driven line-outs in Scotland's 22.
Sexton doubled his and Ireland's tally to open the second quarter, after Joe Schmidt's side were unable to capitalise on advantage for Scotland straying offside in the backline.
Cian Healy was then pinged for collapsing a line-out, and Laidlaw inexplicably punted to touch rather than shoot for goal.
A team turning down three points must then secure five; but Scotland lost the line-out, and with it both scoring chance and initiative.
David Denton cut blind from a scrum in the Ireland 22, but was hauled into touch with the tryline coming squarely into view.
A rushed and dangerous pass from O'Driscoll gifted Scotland plum field position.
Weir faked a drop-goal to keep Scott Johnson's men on the attack, but Ireland forced Max Evans to hold on at the tackle, winning a crucial penalty to relieve the self-induced pressure.
Denton cut away well from another scrum, only for Laidlaw to waste the platform with a loose chip allowing Dave Kearney to clear with ease.
Ireland's left wing then claimed a fine high ball to secure possession again.
Sexton's searing line-break then set Ireland away, and he fired out a pass to Heaslip, who raced down the wing for the left corner, touching down while being hauled into the corner flag.
Referee Craig Joubert referred the decision to Television Match Official Carlo Damasco, and the Italian chalked off the score.
Ireland then turned to the driven line-out to wear Scotland down, with Trimble the ultimate beneficiary.
Heaslip capped a watertight driven line-out with a try as Ireland wrestled control to lead 21-6 at the hour against Scotland in Dublin.
Captain Heaslip bundled in Ireland's second try, registering after having a potential score ruled out before half-time.
Dave Kearney was forced into touch in his own 22 after misreading Weir's punt into the corner.
Toner hit a ruck from the side after the line-out to gift Scotland a penalty, and Laidlaw delivered the points from the tee.
Ireland then built tight phases for the first time in the match, Henry blasting in three times, with O'Mahony, Heaslip and Best all burrowing in too.
Sexton's wayward pass handed Scotland a scrum, but Ireland managed to turn it over.
Luke Marshall found Dave Kearney with a dangerous wide pass, before Ireland won a penalty with Scotland killing the ball.
Ireland punted to the corner, pushing for their second try.
And skipper Heaslip delivered after a fine driven line-out, to make up for the score ruled out in the first half.
The Leinster loose-forward's ninth Ireland try in his 61st cap, converted by Sexton, put Ireland in control.
Sexton, O'Driscoll and Dave Kearney produced a smart midfield combination that almost unlocked Scotland's dogged defence.
Sean Lamont strayed offside to hand Ireland a kickable penalty, and Sexton slotted his third goal of the day.
Weir's astute chip into the left corner forced Ireland into a pressure line-out five metres from their own line as Scotland battled for a foothold.
Replacement Jack McGrath produced a fine turnover to set Ireland back on the front foot, only for Tim Swinson to steal an intercept.
Ireland recovered the ascendancy though, with Sexton's tactical kicking to the fore.
And the hosts added their third try when Rob Kearney capped a neat break from Henry, for a score on his 50th cap.
Sexton converted to stretch Ireland's advantage to 22 points, before making way for Ulster's Paddy Jackson.
Schmidt emptied his bench with the match clearly won, with more than one eye on the visit of Wales to Dublin on Saturday.
Scotland's replacements battled manfully to force some kind of consolation, but in truth the visitors were outclassed.
Jackson's chip almost put Dave Kearney in for a try at the death, but the TMO ruled it out.