Cardiff Blues had enjoyed a strong record against Edinburgh in the PRO12 in recent seasons and went into the game with renewed momentum with four wins in five games in all competitions.
But three yellow cards in Scotland undermined any bid for victory as the region left empty-handed after a frustrating and difficult return to league action.
Edinburgh took the lead early on with a try from Andries Strauss while Chris Dicomidis had a score ruled out for being held up.
Jarrad Hoeata and Ellis Jenkins both spent time in the sin bin in the first half and Sam Hobbs followed in the second as tries from home wingers Dougie Fife and Tom Brown sent Edinburgh clear.
Adam Thomas did grab a consolation in the final ten minutes, but Blues will now need to re-group quickly before a crucial league fixture with Treviso at BT Sport Cardiff Arms Park on Friday.
Tempers flared within the opening minutes with a scuffle off the ball and, following consultation with the TMO, Hoeata was yellow carded as Blues were forced to play a man down early on.
Scotland international Dave Denton looked to make it count immediately with a powerful run stopped by Jenkins before Blues competed aggressively at the breakdown with Manoa Vosawai turning over ball.
A big driving maul then put Edinburgh close and the home side showed patience before they went wide left and used the extra man to send centre Strauss over.
Blues responded with a kick from Gareth Davies, via the right hand post, cutting the lead to four points.
Wing Fish produced a superb kick down the line to force the home side to play from inside their own half.
But Edinburgh duly made big gains downfield and when Josh Turnbull was penalised by referee Leo Colgan for playing the ball on the floor the penalty was confidently sent over by Heathcote as Hoeata returned to the fray.
A neat grubber kick in behind by newcomer Anscombe forced Heathcote to knock on and from the resulting five-metre scrum a penalty for collapsing the set piece was awarded against Rory Sutherland. However, Davies saw his effort cannon back off the post.
Blues looked to play in midfield and keep the ball alive, with Turnbull and Jenkins carrying into heavy traffic. But a penalty against Jenkins for offside, as he looked to disrupt scrum half Sam Hidalgo-Clyne, saw Heathcote knock over his second penalty of the game.
An early change saw Chris Dicomidis replace Hoeata and the lock was straight into the thick of the action.
Blues kicked to the corner, after James Hilterbrand was penalised, and Blues then piled on the pressure. Time and time again the forwards drove for the line before Anscombe opened it up, fed the ball out of the tackle to Davies and he released Dicomidis to run for the whitewash.
The lock crashed over and the TMO was called upon before the decision from the officials was that the ball had been held up. The resulting scrum saw Blues penalised and the chance was gone.
Blues lost a second player to the sin bin, as Jenkins was yellow carded, and Heathcote stepped up to drill over the three points.
Davies nudged over three points of his own just before half-time before a burst from wing Richard Smith saw him hammer past tackles before Fish kicked in behind but could not quite collect the bouncing ball.
Dougie Fife made a darting run at the start of the second half before being hauled down by Fish. And a soft penalty conceded for offside gave Heathcote another kick, but he drifted it wide right.
Blues were disrupted further as Dicomidis was forced off injured, with Turnbull moving into the second row, before yet another offside penalty gave Edinburgh the chance to drive for the line and Fife was duly sent over down the short blindside.
Referee Colgan continued to penalise Blues repeatedly with Sam Hobbs the next to be yellow carded for a penalty in the lead up to the second home try.
A quick tap and go from Vosawai tried to put Blues on the front foot, but stolen ball again set Edinburgh free from deep and a chip in behind from Heathcote forced Gareth Davies to ground the ball for a five-metre scrum for the hosts.
The solid scrum paid dividends as the ball was flung wide and with the extra man it was wing Brown who dotted down for a third home try.
Back up to full numbers, the visiting forwards looked to build some momentum and Macauley Cook went close before spilling the ball.
A classy break from Anscombe put Blues into space and Lewis Jones and Gavin Evans were on his shoulder before Thomas Davies was just short of the line.
Flanker Roddy Grant was yellow carded for Edinburgh, but a missed touch failed to pile on the pressure.
Blues looked to stage a late fight-back and camped on the home line for several minutes with Heathcote yellow carded for a deliberate knock on as Edinburgh were reduced to 13 men.
Walsh sent Thomas over for a consolation try for Blues but there was no further score as Edinburgh wrapped up the points.
08m Strauss t Heathcote c 7-0
13m Davies p 7-3
16m Heathcote p 10-3
25m Heathcote p 13-3
37m Heathcote p 16-3
39m Davies p 16-6
47m Fife t Heathcote c 23-6
59m Brown t 28-6
74m Thomas t Davies c 28-13
Cardiff Blues: 15 Gareth Amscombe, 14 Richard Smith, 13 Adam Thomas, 12 Gavin Evans 9G Smith 78), 11 Dan Fish (T Davies 55), 10 Gareth Davies, 9 Lloyd Williams (L Jones 63), 1 Sam Hobbs (Fish 61 (Walsh 64), 2 Kristian Dacey (R Williams 71), 3 Adam Jones (Andrews 71), 4 Jarrad Hoeata (Dicomidis 27 (Navidi 44), 5 Macauley Cook, 6 Josh Turnbull, 7 Ellis Jenkins, 8 Manoa Vosawai
Replacements: 16 Rhys Williams, 17 Thomas Davies, 18 Scott Andrews, 19 Chris Dicomidis, 20 Josh Navidi, 21 Lewis Jones, 22 Garyn Smith, 23 Geraint Walsh