Leading until late on, Falkirk suffered the agony of crashing to defeat after a couple of late GHK tries.
"For the first 65 minutes we were really, really good,” Falkirk director of rugby Kenny Grieve told the Falkirk Herald. "We coped with the pressure that they applied. They’re a very good side with a lot of ex pros and really good journeyman players.
"But we dominated them at scrum time, we were dominating the collisions and the boys were really performing.
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"The game was quite tight and round about 65 minutes indiscipline cost us. We started giving away cheap penalties, silly penalties which allowed them to then kick themselves into good attacking positions.
"They scored a couple of tries late on to beat us.
"We’re still on a journey with a lot of these lads. We have got some experience in the side that we managed to bring in, but a lot of them are still quite young.
"Just being able to handle that pressure and build pressure at critical times is something that’s missing from a number of them.
"We are disappointed to lose but the players will learn from this defeat and it’s great experience for them.”
Saturday’s game saw GHK score first with a converted try to go 7-0 up after five minutes.
After a period of Falkirk pressure, 1st XV debutant Andy Graham got an interception score under the posts. Duncan Tompkins added the extras for 7-7.
Falkirk then went 14-7 ahead when they were awarded a penalty try after Gregor Ramsay had burst into the 22, fended off three tackles and was stopped short.
Things got better for Falkirk when Blair Watson managed to squeeze over the try line after a barrage from the visitors.
Two GHK tries – one of which was converted and one which wasn’t – levelled the scores at 19-19, before they went on to take the lead from a 5m lineout; 24-19.
But the hosts then received a yellow card and Tompkins kicked a penalty to close the gap to 24-22 at half-time.
Into the second half and GHK landed a converted try after over 15 phases to lead 31-22.
But Ramsay’s second try after great interplay – converted by Tompkins – brought it back to a one score game at 31-29 and secured a bonus point.
A Tompkins penalty then put the visitors ahead at 32-31, but the Glasgow side kicked a penalty of their own to go 34-32 up.
The two late tries then arrived to seal the GHK victory.
The defeat leaves Falkirk sixth in the table with 30 points from 11 matches ahead of their home league game against Dumfries Saints this Saturday, KO 2pm.
"We have now played everybody once so far this season,” Grieve said.
"The game against Dumfries earlier this season was a close encounter which ended in a 29-26 loss down there.
"We are confident that if we can play the game that we want to play and produce the 65 minutes of last week then we should be in a comfortable position to put scores on the board and win.”