Eli Manning, 2004, was effectively drafted by the Giants #1 overall. They just did the trade deal kind of backwards. It's a technicality that he didn't win a SB with his drafting team.
Peyton Manning, 1998.
Drew Bledsoe (1993) technically was the last #1-picked, non-Manning QB to win a SB with his original team, although he was not the starting QB when the Patriots won SB XXXVI. He was the starter when they lost in SB XXXI.
Troy Aikman, taken in 1989, was the last non-Manning QB to go #1 and win it all with his drafting team as the team's starter.
John Elway did the Eli move 20 years prior. Although he played his entire career with the Broncos, reaching 5 SBs and winning 2 after the '97 and '98 seasons, he was picked #1 in 1983 by the Colts, whom he never played for.
Jim Plunkett, #1 in 1971, led the Raiders to 2 SB victories, but he was drafted by the Patriots.
Terry Bradshaw in 1970 is the only other #1 overall QB to win a SB as the starter for his drafting team.
So in 49 years of the SB era, 20 QBs have gone #1 and only 3 have led their drafting team to the pinnacle of the NFL as the starting QB. 3 won SBs with other teams, and 1 was the backup QB for his drafting team when they won a SB.
Other players to be #1 and SB champ with their original team:
Orlando Pace was the most recent non-QB #1 pick to help a team win the SB. STL drafted him in 1997, then won the SB after the 1999 season.
Dallas took Russell Maryland, DT, #1 in 1991. He won 3 SBs over the next 6 years with them.
Dallas also selected Ed 'Too Tall' Jones #1 in 1974. They won SB XII after the '77 season, and he played in 2 other SB's with the Cowboys.
Bubba Smith was #1 in 1967 and played in 2 title games, losing SB III and winning SB V with his drafting team, the Baltimore Colts.
Buck Buchanan, DT, went 1st overall to the AFL's Chiefs in 1963. He was also chosen with pick 265 by the NFL's Packers. He would have gone to a SB with either, as the teams played each other in SB I, but he went with Kansas City and lost that game. A few years later he won a championship with the Chiefs in SB IV. Bonus fun fact: Buchanan was the first black pro-football player ever selected #1 overall.
So by my count, that's 8 overall #1s who helped their drafting team win a SB. 9 if you count Bledsoe.
Honorable Mention:
Jim Grabowski, RB, was picked #1 overall by the Miami Dolphins of the AFL, and #9 by the Packers in 1966. He chose the Packers and won Superbowls I and II with them.
Oh-So-Close (x4):
Bruce Smith, #1 in 1985, was with the Bills during their 4-SB run.