Norrie set the tone when breaking in the opening game of the match before wrapping up the set in 36 minutes.
But the rest of the clay contest unfolded in dramatically contrasting circumstances.
Machac steadied the ship in the second and drew level after saving match point in the tie-break.
The Czech had ample chances to book his spot in the next round but let Norrie off the hook far too many times – taking just one of his 12 break point opportunities.
Norrie found himself a break down and staring at an early exit as Machac served for the match, but he dug deep to break back and held his nerve in the tie-break.
Fellow Briton Katie Boulter was unable to build on her first-round straight-set win over Taylor Townsend as she lost 6-4 6-4 against world number five Jessica Pegula.
Earlier, world number one Jannik Sinner came from a set down to beat France’s Benjamin Bonzi 6-7 (6-8) 6-1 6-4.
It is Sinner’s 18th successive victory after lifting titles at Indian Wells, the Miami Open and the Monte Carlo Masters.
The Italian, who has won 27 ATP Tour events, is seeking his first at the Madrid Open and will face Danish qualifier Elmer Moller in round three.
Russia’s Andrey Rublev, who won the tournament in 2024, fell to a surprise 6-3 6-4 defeat against Czech world number 66 Vit Kopriva.
Croatian Dino Prizmic also pulled off an upset with a 6-4 6-7 (4-7) 7-6 (7-5) win over American world number world number six Ben Shelton.
Kazakh Elena Rybakina, ranked second in the world, overcame a difficult start to beat Romania’s Elena Gabriela Ruse 4-6 6-3 7-5.
World number three Coco Gauff returned from her quarter-final exit at the Stuttgart Open with an emphatic 6-3 6-0 victory against Leolia Jeanjean.