The 19-year-old British number one Murray chalked up a comfortable 6-2, 6-4 victory over qualifier Ivan Navarro Pastor, while 32-year-old Henman rolled back the years with an impressive 7-5, 6-3 victory over another Spaniard Fernando Verdasco.
It was level pegging in the early games, but Henman upped the pressure in the 12th to break serve and take the set. He broke again early in the second and played some fluent tennis as he rattled off an easy victory.
Henman, who lost to Roger Federer in the final of the Japan Open earlier this month, will now meet 14th seeded Spaniard David Ferrer.
"I served fantastically well and at the right times," said a satisfied Henman. "It's a good start for me and good to continue where I left off on Tokyo."
Murray, who has stumbled out at the first hurdle in 10 tournaments this year, avoided the same mistake on his debut appearance in Madrid.
He broke serve in the fourth and eighth games, taking the first set with a delicate lob. He was pushed harder in the second but stepped up a gear at 4-4 and won the match with a classy cross court shot.
Murray, who reached the semi-finals of the Toronto Masters in August, will now meet last year's losing finalist Ivan Ljubicic in the second round.
The Croat, seeded three in Madrid, won his third title of the season with a straight sets victory over Chile's Fernando Gonzalez at the Vienna Open at the weekend.
INTERESTING MATCH-UP
"The best part of his (Ljubicic's) game is obviously his serve. He won in Vienna having not lost a serve," Murray said after the match.
"My return is the best part of my game. It should be an interesting match-up. If I play well I've got a chance but he's playing really good tennis."
Mardy Fish of the U.S. had to battle from a set down to see off Belgium's Xavier Malisse 5-7, 6-3, 7-5 and earn himself a second round match against defending champion Rafael Nadal.
Fellow American Robby Ginepri, who lost to Nadal in last year's semi-final, won a see-saw three setter against Spanish wild card Feliciano Lopez 6-1, 3-6, 6-3 in Monday's late match.
Chile's Nicolas Massu, who lost to Juan Carlos Ferrero in the 2003 final, came from a set down to draw level at a set apiece with former French Open winner Gaston Gaudio but won through after the Argentine retired with stomach problems.
Massu faces a daunting second round clash against top seed and world number one Federer, who is bidding to win his 10th title of the season in Madrid.
"Playing against the best player in the world and on a court which has great memories for me is a great motivation," said the Chilean. "I think I've got a chance."
There was disappointment for another former French Open champion Carlos Moya who was overwhelmed by Juan Ignacio Chela 6-3, 6-2 in his first round match.