Keatley would edge Munster in front with his second penalty, but after missing with a third attempt, Williams made it 6-6 from close to the halfway line.
Leicester took control of the game with a strong finish to the first half as Vereniki Goneva charged over after a driving maul had been halted just short of the line.
Another long spell of pressure produced the second try when Mike Fitzgerald recovered the ball from a lineout to sprint clear, and the conversions from Williams ensured the visitors led by 12 at the break.
Munster made the first breakthrough of the second half when James Cronin pushed the ball over at a ruck, but Keatley was unable to add the conversion.
However, it was Leicester who struck before Ayerza returned to the field as Youngs showed a fine turn of pace to slide over, and a Williams conversion extended the lead to 25-14.