Christie helped the Raptors climb back from a 14-point deficit in regulation and the Magic playing without NBA player of the week Nick Anderson were outscored 12-4 in the final overtime.
Anderson, who had a bad case of the flu, is expected to miss the Magic's game in Charlotte on Saturday night.
Billups converted a three-point play with 10.4 seconds left in the second overtime to force the third extra period, then hit a 3-pointer with 87 seconds left for a 113-107 lead in the third overtime.
Christie added a short jumper for the final margin.
John Wallace had 24 points for Toronto, which won only its fifth road game of the season. Marcus Camby added 14 points and 13 rebounds.
Horace Grant had 20 points and 20 rebounds for the Magic, who blew a four-point lead in the final 26 seconds of the second overtime. Derek Strong had 21 points and 11 rebounds while Derek Harper added 16.
Christie had missed a 12-footer to win it for Toronto at the end of regulation.
Harper's 3-pointer with 29.6 seconds left tied the game at the end of the first overtime. He missed an off-balance layup in the final second that could have won it for the Magic.
Orlando led by nine at halftime and built that lead to as many 14 in the third period before Christie came alive to keep Toronto in the game. He scored Toronto's next nine points, including back-to-back three-pointers to get the Raptors within 71-62 by the end of the quarter.
Toronto chipped away at the deficit in the fourth quarter, eventually cutting it to 84-82 when Christie scored on a putback with 51 seconds to play.
Christie then stole a Gerald Wilkins pass, drove the length of the floor and scored while Wilkins fouled him. He completed the three-point play to make it 85-84 with 33 seconds left in the game.
Danny Schayes made one of two free throws for Orlando with 18.4 seconds to play to tie the game.
Notes: Toronto is 3-1 in overtime games this season while the Magic are 1-1. ... Marcus Camby, Toronto's first pick in the 1996 NBA Draft, was held out of the staring lineup for the third game in a row after starting 38 of 39 games. ... Spud Webb, who was out of the NBA for two years, had his most productive game since signing with the Magic on Sunday. He had eight points and four assists in 18 minutes the longest he has played in his three games.
©1998 The Associated Press. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed