Winston threw for four touchdowns, including a 55-yard score as time expired in the first half on Saturday to help No. 8 Florida State rally from a slow start and beat Boston College 48-34.
BC opened a 14-point lead before the Seminoles scored the next three touchdowns, going ahead for good on the desperation heave to Shaw with 0:00 on the clock.
"I knew I had to pick things up," said Winston, who completed 17-of-28 passes for 330 yards and ran 14 times for 67 more. "We harp on scoring before the end of the half. We did it yet again."
The highest-scoring team in the Atlantic Coast Conference, Florida State (4-0, 2-0 ACC) topped 40 points for the fourth time this season. But they didn't get the blowout they might have expected against a team that won just two games last season and finished last in the conference's Atlantic Division.
"They came out firing," Shaw said. "They came out playing better than we expected. I give them big ups for that."
The Eagles (2-2, 1-1) took a 17-3 lead before Winston threw touchdown passes of 56 and 10 yards to tie it. BC got the ball with 1:49 left in the half and tried to run out the clock, but Florida State coach Jimbo Fisher used his timeouts to get the Seminoles the ball back at their own 40 yard-line with 50 seconds to play.
After Winston was sacked for a 9-yard loss, Devonta Freeman ran for 14 and the Seminoles hurried to the line of scrimmage to get one more play off. Winston received the snap just before time expired, eluded one tackler as he moved up in the pocket and another as he scrambled right, waved to Shaw to go deep and then let the ball go from his own 40 yard-line as he was hit.
Shaw got position on safety Spenser Rositano, jumped at the 5 yard-line, landed at the 1 and fell into the end zone for the touchdown.
"It couldn't have been more on the money," Shaw said. "I knew he could get it there. He can throw it probably on his knees to the end zone."
Fisher said it wasn't a desperation pass as much as a called play, with Shaw running a double fake before heading to the end zone. Shaw said he didn't have any thought about breaking off his route to bail out Winston as he scrambled out of trouble in the backfield.
"There was no time left," he said. "I was going to the end zone."
For Boston College, it was a big swing in emotions after taking a lead, then almost getting back to the locker room with the game tied 17-all. The Seminoles barely got the snap off in time, BC coach Steve Addazio said, and twice the Eagles seemed to have him trapped in the pocket.
"It was close," he said. "Then I got caught up with the fact that I thought we had him on the ground -- twice. I'd seen that same thing happen on tape, where he'd spun out of a couple of tackles and threw a dart down field. Give him credit. He made a great football play."
Winston added a fourth TD pass in the third quarter, a 10-yard score to Rashad Greene that made it 31-20.
Chase Rettig completed 19-of-29 passes for 197 yards and a career-high four touchdowns for BC (2-2, 1-1), including a 52-yard score to Myles Willis that cut a three-score lead to 38-27. But the Eagles quarterback also threw an interception early in the fourth that T.J. Williams returned 20 yards for a touchdown to make it 48-27.
BC took the lead on its opening possession when Spiffy Evans returned a punt 19 yards to the Florida State 36 and Williams ran six times in seven plays before Rettig hit C.J. Parsons for a 6-yard TD pass. The Eagles made it 14-3 when Rettig connected with Jake Sinkovec for a 3-yard score.
It was 17-3 when Winston hit Greene for a 56-yard score, then went 4-for-4 and ran twice for 27 more yards on a seven-play, 80-yard drive that tied the game.