The goal, Davis' first-ever in the competition, gives Houston a slight advantage heading into the second leg next Wednesday in Torreon, Mexico.
An intriguing and lively contest between teams at two very different points in the domestic calendars entertained the fans at BBVA Compass Stadium. Santos has already played nine Liga MX games, while the host opened its Major League Soccer season with a 2-0 victory over on Saturday.
The distinction in fitness and fluidity between the sides was negligible, with the pair both appearing to be in mid-year form.
Houston nearly took a 19th minute lead, when a header by the imposing Bobby Boswell off a corner kick by Oscar Boniek Garcia was blocked at the line.
Seconds later, an errant kick by Dynamo goalkeeper Tally Hall was gathered by Santos' Herculez Gomez. The U.S. international found a streaking Carlos Darwin Quintero, but the Colombian's attempt from 18-yards flew over the crossbar.
In the 22nd minute, Giles Barnes failed to locate the net with Santos goalkeeper Oswaldo Sanchez far out of his box after a mistimed challenge.
Shortly before the half, Houston's Kofi Sarkodie recovered quickly from a mistake in his own box to thwart the best chance Santos had in the opening 45 minutes.
In the early stages of the second stanza, Oribe Peralta - named the best player of last season's Champions League - fired a shot right at Hall following a neat interchange between Quintero and Gomez.
Santos couldn't capitalize on another Hall miscue in the 64th, however, Sanchez did make a save from an additional Boswell header.
Then, in the 89th minute, Davis - a second half substitute - submitted his heroic moment. Corey Ashe dribbled down the left and slipped a pass to the Dynamo midfield maestro. Davis took a touch just inside the box and placed a left-footed effort past a diving Sanchez for the game winner.
Houston snapped Santos' five-game Champions League winning streak, while posting its first win over a Mexican side in six competition outings.