"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.
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'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.
"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."
Ryan threw for 231 yards, and in the absence of Steven Jackson one of several sidelined Falcons Jacquizz Rodgers and Jason Snelling combined for 139 yards rushing.