"We have a special team, a team with a lot of talent," receiver Brian Hartline said. "We're going to continue to improve. The sky's the limit."
Miami fell behind 10-0 and 20-10 and trailed much of the game, but nonetheless improved to 3-0 for the first time since 2002. The injury-plagued Falcons, who were one play from the Super Bowl last season, fell to 1-2.
An enthusiastic crowd of 70,660 reflected the buzz building about the surprising Dolphins, who are trying to end a streak of four consecutive losing seasons.
"We've got a damned good football team," owner Stephen Ross said. "The quarterback is maturing."
The last-minute comeback victory was the first for Miami's second-year QB. Tannehill finished 24 for 35 for 236 yards and two scores. He shook off two turnovers and five sacks, and was at his best at the end of each half.
On the final drive, Tannehill went 8 for 11 for 69 yards, with two of the incompletions because of dropped passes. The winner came on second down, when Tannehill lobbed his throw over linebacker Stephen Nicholas to Sims, a backup tight end.
Sims scored his first NFL touchdown by pulling the ball in with his left hand because his right arm was tangled up with Nicholas.
"It was awesome. It was amazing," Sims said. "The defender grabbed one hand, so I had no choice but to use the other one."
Tannehill suggested the play to his coaches during a timeout just beforehand.
"It's a play we know very well," he said. "It's either going to be a touchdown or a throwaway. We had a lot of confidence in that play."
Tannehill's other scoring pass was an 18-yarder to Hartline to make it 20-20 late in the third quarter. Tannehill also moved Miami 60 yards in the final 2:04 of the first half for a field goal on the final play.
The victory allowed Miami to keep pace with New England atop the AFC East.
"It was a good step forward," Hartline said. "Expectations will continue to rise, and high expectations bring out the best in you. We welcome it."
The Falcons believe they could be special, too, but they lost despite advantages of 24-16 in first downs, 377-285 in yards and 5-0 in sacks.
"It was a game where we had a ton of opportunities," coach Mike Smith said. "We need to be more efficient in the red zone on both sides of the ball. We had way too many field goals."
Lamar Miller had a 49-yard run to set up the Dolphins' first touchdown, but they generated little running attack otherwise. Special teams made a pivotal contribution when rookie Don Jones' jarring tackle on punt returner Harry Douglas forced a fumble the Dolphins recovered to set up a touchdown.
"That was the biggest play of the game," Miami receiver Mike Wallace said.
The Dolphins lost defensive end Cameron Wake (knee) and linebacker Koa Misi (shoulder) to injuries, but the defense dug in after a slow start. Consecutive possessions by Atlanta in the second half netted a total of 10 yards, which allowed the Dolphins to rally.
"Guys had to step up," said first-round draft pick Dion Jordan, who came off the bench to see his most extensive action of the season. "We stuck together and it came up big."
The Falcons scored on their first three series, and at one point the Dolphins trailed in total yards by 127-3. Atlanta had the ball for nearly 23 minutes in the first half but was up only 13-10 at halftime.
Ryan led drives of 14, 11 and 13 plays on the Falcons' first three possessions, but twice they stalled and settled for field goals.
"We had a lot of opportunities to finish the game," Ryan said. "We just didn't get it done in the red zone. We have to find a way to punch it in."
The Dolphins have scored at least 20 points in each game, the first time they've done that since 2002. ... Atlanta's Matt Bosher, leading the league in punting, averaged 33.0 yards on two kicks. ... Osi Umenyiora had his first two sacks for Atlanta. ... Miami rookie Caleb Sturgis kicked field goals of 46 and 50 yards and is 6 for 6 this year.
Tom Brady has won his appeal against the NFL, with U.S. District Judge Richard M. Berman criticizing commissioner Roger Goodell for dispensing "his own brand of industrial justice."
Commissioner Roger Goodell will skip the NFL season opener between the Steelers and Patriots next Thursday in Foxborough, a league spokesman told Fox Sports.
Las Vegas sportsbooks reacted to news of Tom Brady's appeal win by reinstalling the Patriots as the co-favorites in the AFC and bumping up the point spread in next week's season opener.
The decision to vacate Tom Brady's suspension could impact how many games Greg Hardy plays for the Cowboys in 2015, as he will confer with the NLFPA whether to challenge his own ban in court.
Some of the former New Orleans Saints affected by the harsh sanctions of Bountygate three years ago were glad to see NFL commissioner Roger Goodell take another hit on Thursday.
A jury says former Indianapolis Colts linebacker Josh McNary is not guilty of forcing a woman he met in a bar to have sex.