Josh Smith finished with 19 points for the Hawks, who took their sixth straight home victory against Orlando.
Atlanta moved 1½ games ahead of New Jersey, which lost to Philadelphia 91-87, for the eighth and final playoff spot in the Eastern Conference.
"It's not comforting," Johnson said. "We have no room for error."
The Magic, who dropped to 22-2 on the road when entering the fourth quarter with a lead, had won seven of eight overall.
Bibby caused matchup problems for the Magic and had a game-high eight assists.
Dribbling from the perimeter into the top of the lane, Bibby drew Orlando center Dwight Howard and another defender toward him. Al Horford cut in behind Howard to catch Bibby's bounce pass and hit a runner that gave the Hawks a 91-85 lead with 2:04 remaining.
"Joe refused to let us lose, he and Bibby both," Hawks coach Mike Woodson said. "I wanted to rest both of them, but it's time now. We just have to keep pushing and scraping and scratching to do whatever we can to win games. You can't afford to lose now."
Johnson hit five of seven shots from the field in the third quarter, and finished off the win with two free throws in the final seconds.
Rashard Lewis gave the Magic their last lead with a 3-pointer that made it 76-74 midway through the fourth.
The Magic are now 46-26, third-best in the East, and despite the loss, they still lead fourth-place Cleveland by five games.
Howard, who leads the Magic with a 21.1 scoring average, took a season-low five shots and finished with just 11 points. He was just 5-for-11 on free throws.
"I don't necessarily think it was just them," Howard said of the Hawks. "I think it was us. We didn't play as well as we needed to play. We didn't come out with a lot of energy. We didn't play as focused as we normally play. They came out hard and aggressive."
Having played nine games in 15 days, Howard indicated that fatigue affected the Magic considerably.
"Oh, man, I think it was," Howard said. "I think it was."
Bibby, acquired in a trade last month with Sacramento, showed why the Hawks valued his experience at point guard.
His 19-foot jumper with 48.7 seconds remaining made it 93-87, but the play unfolded because Bibby worked so well with Johnson.
"Mike controlled the tempo," Hawks guard Josh Childress said. "The fourth quarter was pretty much his, and that's the reason we won."
Seeing his teammate drive into the right side of the lane and draw a second defender, Bibby moved quickly out to the left wing with Nelson between him and the basket. The shot clock was winding down when Bibby took Johnson's pass, dribbled to his left and used a stutter-step to make Nelson jump too soon.
When Bibby's shot rattled in, Orlando coach Stan Van Gundy called timeout to remind his players that the performance was unacceptable, even if they erased a 14-point deficit in the first half.
"So what? We lost," Van Gundy said. "I'm not patting them on the back for losing. To a sub-.500 team."
Horford had 10 rebounds. ... The Magic have 23 road wins, third in the NBA behind the Celtics and Lakers. Orlando's next victory on the road will set a single-season franchise high. ... Atlanta is now 8-11 since trading for Bibby. ... Overall, the Magic dropped to 45-5 when entering the fourth with a lead.