Portland, making its first playoff appearance since 2011, had been eliminated in the first round in its last six trips, and hadn't advanced to the second round since 2000.
Dwight Howard made four consecutive free throws to tie the game at 91 before making a bank shot that put Houston ahead with 3:04 to go. Robin Lopez scored on a tip shot to give the Blazers a 94-93 lead with 2:15 left, setting up a wild finish.
Howard's layup and foul shot put the Rockets back up 96-94 before Nicolas Batum's fadeaway jumper tied it again with 39.9 seconds left. Chandler Parsons scored on a reverse layup with 0.9 seconds left to give Houston the lead.
Lillard prevented the series from going back to Houston with his 3-pointer as time ran out. He finished with 25 points.
Portland will face the winner of the series between San Antonio and Dallas. The Mavericks forced a deciding Game 7 with a 113-111 victory over the Spurs on Friday night. There are five first-round playoff series that have gone to seven games.
LaMarcus Aldridge led the Blazers with 30 points and 13 rebounds.
After losing the opening two games in Houston, then claiming one of a pair in Portland, the Rockets staved off elimination with a 108-98 victory on Wednesday night to narrow Portland's advantage in the series to 3-2. Howard had 22 points and 14 rebounds, and Jeremy Lin redeemed himself from a poor performance in Game 4 with 21 points off the bench.
James Harden led the Rockets on Friday night with 34 points. Howard had 26 points and 11 rebounds as Houston - which earned the nickname "Clutch City" back in the mid-1990s when the team won back-to-back NBA championships - failed to become the ninth team to come back from a 3-1 deficit in a best-of-seven playoff series and advance.
The Rockets went up 27-17 in the first quarter, led by Harden with 13 points. Portland closed the gap with an 11-2 run to close out the quarter, pulling within 29-28. The Blazers went ahead 34-32 on Thomas Robinson's 16-foot jumper, but the lead was short-lived.
Lillard's 3-pointer put Portland in front 54-50 with about two minutes left, but the Rockets led 58-56 at the half. Harden had 24 points and was 10-for-10 from the free-throw line in the half.
Howard, who collected three fouls in the first half, dunked to put the Rockets up 66-58 in the third quarter. Houston held on to the lead but Aldridge narrowed it to 77-66 with a dunk late in the period.
Nicolas Batum made a 3-pointer and Robinson had back-to-back baskets early in the fourth to give the Blazers an 85-81 lead.
Howard traded shots with Lopez, who put Portland ahead 89-87 with six minutes left. He added free throws to cushion the lead to 91-87.
Aldridge helped the Blazers steal the first two games in Houston, scoring 46 points in a 122-120 overtime Game 1 win, then 43 in a 112-105 victory. The Rockets claimed the first of two in Portland, 121-116 in overtime, led by Harden with 37 points. The series' third OT game ended in a 123-120 Blazers win.
NOTES: The Blazers honored former coach Jack Ramsay, who led the team to its lone NBA championship in 1977, in a moving pregame ceremony. Ramsay died Monday after a long battle with cancer in Florida. ... A fan at the game held a sign reading: "Play like it's 1977." ... Lillard has multiple 3-pointers in six straight playoff games, a franchise record.