Connolly scored on a power play 1:04 into overtime -- his second goal of the game -- and Miller made 33 saves to help the surging Sabres beat Toronto 3-2 on Friday night.
Miller improved to 8-0-1, and the Sabres (8-1-1) won their fourth straight game to match their second-best start after 10 games since entering the NHL in 1970.
Toronto, an NHL-worst 1-7-3, was all over Buffalo in the third period, with Miller preserving the Sabres slim 2-1 until the final minute when Mikhail Grabovski deflected Tomas Kaberle's shot to tie the game.
The short-handed goal came with Toronto goalie Jonas Gustavsson off for an extra attacker with 37.3 seconds left in regulation, and was upheld after a short review showed Grabovski making contact with the puck with his stick just below the crossbar.
The Maple Leafs finished with a 16-5 shots edge in the third.
"That's who he is. He's a battler and the backbone of our team," Buffalo captain Craig Rivet said about Miller. "We played against a feisty team that wanted a win, and he stood on his head a couple times to help us out. He was the main reason we got the win."
Drew Stafford also scored for Buffalo, while Connolly netted the winner when his slap shot from the middle of the blue line eluded Gustavsson.
"It's a huge win for us," Connolly said. "That's one of our best wins of the year. They controlled the play for a lot of the game, but to battle through and win, those games are important. You have to be able to win games when there's not a lot going on."
Roles seemed to be reversed in a game that featured two teams on opposite ends of the standings. But Miller quickly became the difference when he turned aside 13 shots in the first 10 minutes of the third period.
"[Miller] grabbed the bull by the horns and did his thing, which was great to see," Sabres coach Lindy Ruff said. "Our energy level wasn't very good. They came out with the all-out press, and you have to give them credit."
Ian White also scored for Toronto, and Gustavsson made 25 saves. The Maple Leafs have secured at least a point in three straight games.
"We played a solid game but sometimes it's not enough," said Kaberle, who had a pair of assists. "We did a good job going to the net in the third. The first few games in the season we were just standing around and waiting for good plays and empty nets that weren't there."
Stafford opened the scoring with 4:49 left in the first period, shoveling home a loose puck in the crease with Buffalo on a two-man advantage.
White tied it with a slap shot from the right point that got past Miller with just over a minute to go in the first.
Connolly made it 2-1 midway through the second period when he beat Gustavsson on the stick side with a shot while on a breakaway.
"We're doing a lot of things well," Toronto coach Ron Wilson said. "If we keep doing this, the results are going to be there. We have to keep persevering."
The Sabres are 9-2 in their last 11 games against Toronto, including winning six straight. ... Buffalo D Toni Lydman missed his third straight game because of a groin injury. ... The Maple Leafs, who came in with the best power-play unit in the league, finished 0-for-4 with the man advantage. ... Toronto plays at Montreal on Saturday to complete a five-game trip. ... The Maple Leafs have led after two periods just once in 11 games. ... The 1975-76 Sabres also were 8-1-1 after 10 games.