Eric Elwood, in his last season in charge of the Irish outfit, had challenged his side to improve on last season's eighth place finish and this result sees them continue to close the gap on Treviso in seventh.
Four wins in their last five proves Connacht are a side in form while Edinburgh's run of two wins in a row is now at an end despite seeing Laidlaw slot 24 points with the boot.
Edinburgh started the brighter of the two sides but it was Connacht who hit the front early on, Dan Parks slotting over a penalty after the Scottish side did not release in front of their own posts.
But it did not take long for the hosts to respond as Laidlaw made no mistake with a penalty of his own to level it up at 3-3.
The hosts were shading the early exchanges and another nerveless kick from Laidlaw put Edinburgh in front on the quarter-hour mark.
But not for nothing have Connacht won three of their last four in the league and they looked on the verge of the first try of the game as the game reached the 20th minute.
A series of drives on the try line brought Elwood's side close but when the ball went wide it was knocked on and the chance was gone.
And no sooner had Connacht missed their chance than they found themselves 12-3 down, two Laidlaw penalties either side of a fine Tim Visser break putting the Irish region on the back foot.
As half time approached Elwood's side did not deserve to be nine points adrift and they went over for the first try of the game after a quick counter-attack.
Mick Kearney found himself in a two-on-one situation and the flanker eased over, Parks adding the extras to reduce their arrears to only two points at 12-10.
Before half-time the Gunners came close to their first try when Visser hared after a clever Matt Scott kick and when Connacht infringed at the break down Laidlaw had his fifth penalty of the half for a 15-10 lead.
That looked like being the end of the scoring for the first period but despite the sin-binning of Parks there was still time for Gavin Duffy to step up and slot a long range penalty and send Connacht in only 13-15 down.
With Stevie Scott and Hodge's words ringing in their ears it was the Gunners who started brighter and a fine Matt Scott break almost brought their first try.
Connacht went off their feet but Laidlaw missed for the first time on the night to keep the gap at two points.
Elwood's side came straight back and after some impressive Scottish defence on their own five-metre line they eventually conceded a penalty which Parks slotted to put Connacht 16-15 in front.
And with the hour-mark approaching Parks was on target again as the visitors began to turn the screw with their direct approach and make it 19-15.
Edinburgh were soon back on the front foot and Ronan Loughney conceded a penalty that saw him get Connacht's second yellow card of the night.
Laidlaw made no mistake with the penalty and then added another on 60 minutes as the hosts made their one man advantage count to take a 21-19 lead.
But despite still being down to 14 men Parks soon had Connacht back in front, keeping his goal to slot a drop goal and make it 22-21.
Fetu'u Vainikolo then came close to a try with a searing break but Netani Talei was alert with a last-ditch tackle to haul him down and keep Edinburgh in the game as Loughney returned to the field.
But no sooner had the prop retaken the field than his side were back behind as Laidlaw was again on target from the kicking tee.
Defending a narrow lead the indiscipline shown by Edinburgh will have frustrated Scott and Hodge and when they conceded another penalty, admittedly wide out on their own 22, they would have been fearing the worst.
But Parks picked a bad time to miss his first kick of the night, his effort sliding wide as Edinburgh looked set to hold on.
But in the closing stages a sweeping break from Connacht brought them right up to the try-line and Henshaw burrowed over for the try, Parks adding the extras and a penalty with the last kick of the game to deny Edinburgh a losing bonus point.
|Edinburgh Rugby Score Card|
|Connacht Rugby Score Card|