In driving rain and snow at frigid Foxborough, Maye overcame a torrid outing that included coughing up four fumbles and being sacked five times to take the Patriots into next Sunday's AFC Championship game where they will face the Denver Broncos.
The Patriots quarterback delivered in the key moments, notably connecting with Boutte for an extraordinary fourth quarter touchdown that saw the New England receiver dive at full stretch to snag a one-handed catch in the end zone that effectively sealed victory.
Maye's roller-coaster game was surpassed by Houston counterpart C.J. Stroud, who tossed four interceptions in a nightmare first-half display that helped the Patriots take what would turn out to be a decisive lead.
The Patriots' victory was further proof of the NFL giants' renaissance under new head coach Mike Vrabel, who led New England to a 14-3 record in his first campaign in charge after the six-time Super Bowl champions had gone 4-13 in the previous two seasons.
"Just proud of the guys - we battled the elements and our defense played a hell of a game," Maye said. "I just kept reminding the guys, we're still ahead on the scoreboard and that's what matters. And we made enough plays to win."
The Patriots took the lead when Maye picked out DeMario Douglas with a 28-yard pass on a fourth down conversion to make it 7-0.
The Texans responded with a 25-yard Ka'imi Fairbairn field goal to cut the lead to four points, and then capitalized on a costly Maye fumble to score a touchdown after Stroud picked out Christian Kirk in the end zone.
But that proved to be the highpoint of a miserable first half for Stroud, whose passing game disintegrated in the face of a swarming Patriots defense and the challenging weather conditions.
Stroud gave up a pick six soon after Houston took the lead, his pass to Xavier Hutchinson being grabbed by Marcus Jones who sprinted 26 yards into the end zone for a 14-10 lead.
Stroud then tossed another interception on Houston's very next possession, although the Patriots were unable to take full advantage and punted away.
New England did not have long to wait though before they extended their lead, Maye aiming a dart to receiver Stefon Diggs to help take the score to 21-10.
The Texans regrouped in the second half and cut the Patriots lead to five points after back-to-back Fairbairn field goals.
New England though extended their lead to 12 points with Boutte's touchdown and the defense held firm to clinch victory.