McGrady, a seven-time All-Star who has never advanced past the first round in the postseason in six attempts, tried his hardest with 29 points, matching the total of teammate Yao Ming, to get his side to an 88-83 lead with 6:09 to play after being 16 points down.
With Houston trailing 75-67 after three quarters, Yao led a 7-0 run to open the fourth period, but the Rockets did not pull level until guard Rafer Alston hit a 3-pointer to tie the game at 80-80 with 8:38 to play.
Four free throws by Yao and back-to-back buckets by McGrady helped the Rockets stretch the advantage to 88-83, only to see the home side storm back with an 8-0 run as Boozer sandwiched five points around a 3-pointer by Andrei Kirilenko for a 91-88 lead with 4:15 remaining.
After Yao tied things again with a three-point play, his opposing centre Mehmet Okur sank a 3-pointer for a 94-91 lead with 3:23 left, and then buried an almost identical maximum to make it 99-95 with 1:16 to go.
Two free throws by Yao cut the deficit to 99-97 with 57 seconds left, but Boozer replied with two of his own to take a four-point lead with 19.9 seconds to play.
McGrady drove for an uncontested layup with 9.9 seconds left to get the deficit to 101-99, but the Rockets inexplicably failed to foul until 1.7 seconds were left, before Kirilenko nailed both foul shots to clinch it as the home team failed to win for the first time in the series.
The fifth-seeded Jazz moved through to the second round for the first time since 2002 and will play eighth-seeded surprise packets the Golden State Warriors in the conference semi-finals that begin in Utah on Monday.
In the other Western Conference semi-finals the second-seeded Phoenix Suns take on the third-seeded San Antonio Spurs.
Meanwhile in the opening match of the Eastern Conference semis, the Detroit Pistons thrashed the Chicago Bulls 95-69 in Detroit to lead their seven-game series 1-0, with guards Chauncey Billups and Richard Hamilton each scoring 20 points.
In the other Eastern Conference semi-final match-up, the Cleveland Cavaliers take on the New Jersey Nets in a best-of-seven series starting in Cleveland on Sunday.