Earlier Ross Samson's try, coupled with the nerveless kicking from Tom Homer, looked to have put the Exiles well on route to victory.
But Bradley's men held firm and got themselves back into the game through man-of-the-match Stuart McInally's try before Laidlaw's late heroics.
"That was a very tight battle and could have gone either way. But we are delighted with the result," said Bradley.
"We defended very well in that last period - it was outstanding defence with no penalty conceded and no try.
"Those sort of things make tournaments and they make seasons. We were 19-10 down at one stage and had to fight hard. We did that and we kept our shape and we got the win in the end.
"We played some very nice rugby when allowed to, and I was impressed by our discipline when Irish put on the pressure at the end.
"No penalty conceded while defending phase after phase when one goal was all they needed."
England international centre Shontayne Hape was at fault for Laidlaw's late penalty as his apparent retaliation to a forearm from Sean Cox earned him a yellow card.
But Bradley refused to condemn either player after a vicious physical battle from the first minute to the last.
"Rugby is a physical game. You have incidents on the pitch," Bradley added.
"As a coach I look at it in terms of results. If there was a running battle there was a running battle, Cox and Hape were very competitive out there but rugby is a very physical game."
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