The Heat looked to be coasting to victory when they led by as many as 13 points in the fourth quarter, but Brooklyn refused to lie down as they reduced the deficit to 93-91 with 35 seconds remaining.
Luol Deng was successful with two free throws with 11 seconds on the clock and there was still time for Jarrett Jack to miss a lay-up before Johnson failed to convert both free throws.
Johnson, who was also off target with an attempt at a three-pointer, was encouraged by the way his side fought back and is eager to ensure there is no repeat of the mistakes they made late on.
He said: "I thought we'd done a great job in fighting back, had a couple of threes go down - two points [behind] with 35 seconds left, it was a tough call to make a time like that to given them two free throws, but we turned the ball over, made too many unfortunate turnovers down the stretch.
"Very uncharacteristic for us. So those are things we've got to look out for. Just take care of the ball and value the basketball when it's late in the game."
Mason Plumlee top scored with 21 points for the Nets and he is confident Lionel Hollins' side will be more clinical against the Toronto Raptors on Wednesday.
"[on missing shots] At home maybe but we've just come off Charlotte where we shot the ball great. We missed shots tonight but guys are going to hit.
"We have too many guys who are good shooters but when we've had bad shooting nights like we did against Philly we've come back the next night and shot the ball great, so hopefully that'll be the case tomorrow."