Kyrgios, currently without a coach and to whom Stepanek has already passed on playing tips, kept both his power game and his notorious temper in check in their first-round clash.
He controlled much of the match as he moved Stepanek around with three quarter-pace groundstrokes fired deep into the corners.
But after faltering in a marathon third set tiebreak in which the Czech saved two match points, the Australian admitted he was relieved to come through in four.
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"That's a nightmare first round for anyone at Wimbledon. No one wants to play Radek here," he told reporters after his 6-4 6-3 6-7(9) 6-1 win.
At 37, wildcard Stepanek was bidding to become the oldest man to win a grand slam singles match since Jimmy Connors reached the second round of the U.S. Open in 1992.
Playing to a soundtrack of encouragement from the Antipodean glee club of green and gold-clad fans that follow Australian players around the circuit, Kyrgios won the first set with a service break in the final game.
Also watched by a more illustrious compatriot, octogenarian three-times grand slam singles winner Neale Fraser, he then had the chance to wrap things up in straight sets.
But the Czech broke serve at 4-5 and took the breaker by 11 points to nine, luring Kyrgios to the net with one of many silky dropshots before passing him with a crosscourt forehand.
Tiring the quicker of the two, Stepanek failed to repeat his first-round heroics of the French Open, where he went down to world number two Andy Murray in five sets, and Kyrgios stormed through the fourth set with two further breaks.
Of a prospective future coaching partnership with Stepanek, the Australian said: "When he stops, he might step into a role where I might bring him along to tournaments... I'm more than willing to give it a go with a guy like that."
The Australian next plays unseeded German Dustin Brown in a clash sure to grab its share of the media spotlight on Thursday, the mercurial German having knocked out Serbia's Dusan Lajovic in five quickfire sets.
"When you're playing a guy like Dustin, you hope it's not a day when he's feeling it... We all know what he can do," Kyrgios said.
(Editing by Ed Osmond)
