But missed foul shots, failed bounce passes through the legs and the inability to finish a defensive possession doomed them, and the Cavaliers finished them off 128-120.
James Harden made a little history, becoming the franchise's first player to have at least 40 points, 15 assists and five rebounds in a game, according to the Elias Sports Bureau.
But his three missed free throws in the fourth quarter left him shaking his head.
Harden went 4-of-7 from the line in the final quarter, critical misses that could have placed more pressure on the Cavs down the stretch.
"It's on me," said Harden, who scored 41 points. "I missed [three] foul shots tonight. Can't happen. I get there pretty often. A couple missed foul shots, missed opportunities to make a couple of 3s, that's the ball game right there."
With the Rockets down seven, Harden missed two foul shots with 3:19 left. Houston kept fighting, and after trailing 119-114 with 1:26 left, Harden drew three foul shots. He made just two.
If he converts all three, it's a two-point game, and maybe the Rockets don't pressure Kyrie Irving in the frontcourt, something coach Mike D'Antoni didn't want to do.
But they did, and when doubled, Irving is dangerous. Irving passed out of the double-team, and the ball eventually made its way to J.R. Smith alone in the corner, opposite of the Rockets' bench.
"We were scrambling a little bit," D'Antoni said.
Smith swished the 3, pushing the Cavs' lead to 122-116, and LeBron James and the Cavs were soon heading next door to watch the Indians in Game 6 of the World Series.
Late turnovers by Houston also compounded their problems.
On consecutive possessions midway through the fourth, Clint Capela and Harden committed the same type of turnover, contributing to the Rockets' deficit increasing from four to 10.
Capela held the ball near the foul line and tried a bounce pass between defenders to Harden. It's a pass Capela isn't used to making, and the turnover led to a Mike Dunleavy 3, pushing the Cavs' lead to seven.
On the Rockets' next possession, Harden tried the same pass, this time to Capela, and it was stolen by Kevin Love. Love was fouled by Trevor Ariza attempting a 3 in front of the Rockets' bench. He made all three of his foul shots, pushing the Cavs to a 105-95 advantage with 6:06 left.
"That's Clint's growth," D'Antoni said of the turnovers. "He'll get better at it."
Still, the Rockets didn't approach this contest as if they needed to steal a victory from the defending champs. This new Rockets' team has a mindset that it belongs with the elite teams.
Victories against elite teams will only build confidence.
After losing opening night in Los Angeles to the Lakers, the Rockets won two solid games against the Dallas Mavericks.
They opened this five-game road trip against the defending champs looking to make a statement of sorts.
"[Tuesday] was a good test for us," Harden said. "It didn't end the way we wanted it to, playing against the defending champions on their home court, but we can take some really good things out of that game. Some mental mistakes, some calls didn't go our way, but overall we have an opportunity to get better tomorrow."
The Rockets felt like they let a victory escape them Tuesday, but Cleveland took the game over by making its last seven shots over the final 6:45. Championship teams accomplish those things.
"I think we got a lot of good looks, and we just didn't make shots," said Ariza, who took five stitches under his right eye following a head butt with Tristan Thompson. "They slowed it down, they executed or they got to the free throw line."