Ceci, who turns 20 on Saturday, beat former Senators goalie Brian Elliott with a shot from the right point for the game-winner, which came in his third NHL game since being recalled from Binghamton of the American Hockey League on Thursday.
"It was just a really special moment for me," Ceci said. "To have my family and friends in the building, and for it to go in in overtime, and to win the game against a great team, it was pretty crazy the way things happened, and I'm happy the way they did."
Ottawa backup goalie Robin Lehner stopped 30 of 32 shots.
"I was really happy," Lehner said. "The first thing, to see him score that one, and the second thing, to see me not going to the shootout. It felt pretty good, to be honest with you."
Bobby Ryan had two points for Ottawa, including his team-leading 16th goal at 5:44 of the third to tie the game at 2-2.
"I think if you could have written a script for us, that's the way it would have gone," Ryan said. "Cody's played a good few games for us."
Ryan, who left the game briefly late in the second after crashing head first into the end boards, also assisted on Jean-Gabriel Pageau's first-period goal, which gave the Senators a 1-0 lead.
"I just kind of jammed my shoulder," Ryan said. "You get that tingling sensation right down to your fingertips and you felt like you're on fire, but other than that, it was fine after five or six minutes and a little ice, and nothing to write home about."
Lehner has seen action in two straight games after relieving Craig Anderson on Saturday after Anderson allowed two goals on the first four shots he faced in the Senators' 5-2 loss to the Los Angeles Kings.
"After the L.A. game, the morale was a little low here the last couple of days," Ryan said. "We've had a lot of meetings on how to correct things and how to get going, and we've always come back to the start of the game, so for us to come out and come back in here after 20 (minutes) with the lead, it meant a lot."
Ottawa completed a three-game homestand at 2-1-0. The Senators play the New Jersey Devils on Wednesday.
Chris Stewart scored goals 2:57 apart late in the second period to give St. Louis a 2-1 lead.
Stewart, who has four goals in three games, drew the Blues even at 1-1 at 16:50. He put St. Louis ahead with his ninth goal with 12.7 seconds left in the middle period.
Lehner slid to his right and stacked his pads to make a sensational save on Alexander Steen's shot moments before Stewart's first goal.
"I was a little bit too committed to the guy with the puck, and when he passed it over, it was too far away, so instead of reaching, I just did something and I was fortunate it hit my pad," Lehner said.
Blues coach Ken Hitchcock was impressed with his team's effort in battling back from the early 1-0 deficit after Ottawa held a 10-7 shot advantage in the first.
"They got their (butts) handed to them by Los Angeles, so they were [angry]," Blues coach Ken Hitchcock said. "They were mad and they played really hard in the first period, but I thought the way we played in the second period was great. The second period was one of the best road periods we've played all year. We really took it to them and really hemmed them in, and at times overwhelmed them with our forecheck, so I was really happy. I was happy with the point.
"You know, disappointed that we lost it the way we did, it was too bad. [We] made a few errors on the 4-on-4 stuff, but overall the way we played in the second and third period was pretty impressive. We were down three, four top forwards, that's pretty impressive."
Blues captain David Backes left the game during the second period with an upper-body injury and did not return.
Backes skated slowly off the ice and went to the dressing room with 2:49 left in the second after he collided with Ottawa's Colin Greening on an open-ice hit.
Backes had seven hits and won 71 percent of faceoffs in 11:20 in ice time before he left the game. He has 16 goals and 14 assists this season, second on the team behind linemate Steen (35) in points.
"We'll let you all know on all the injuries [Tuesday]," Hitchcock said.
The Blues, who face the San Jose Sharks on Tuesday, ended a two-game road trip that began Saturday with a 4-3 overtime win against the Columbus Blue Jackets.
"That's not the way you want to lose a game," said Elliott, who made 29 saves in the loss. "We all kind of made mistakes tonight and I thought we battled hard to get back, and we scored a couple of big goals there, especially the one at the end of the period, but it's a tough way to lose. We got a point out of it here, and we've got to answer right back San Jose at home [Tuesday]."