The 19-year-old, who is ranked 144 in the world, has caused one of the early shocks of the tournament after knocking out the Frenchman.
Gasquet clinched an early break of serve in the first set to keep himself on top and eventually take a one-set lead on the grass court.
The Frenchman continued his dominance by grabbing another early break in the second set, but Kyrgios showed signs of a comeback by earning a break of his own to level on 5-5.
A tie-break was needed to decide the set, but it was Gasquet who managed to edge it 7-4 to grab a two-set lead at the All England Club.
The third set was a tight affair between the two competitors until Kyrgios came through the seventh game to take a vital break after it was brought to deuce twice.
After getting a set back, Kyrgios continued to push Gasquet to the limit and eventually managed to hold on to clinch the fourth set after securing a late break of serve.
The fifth and final set was a close contest once again as Kyrgios relinquished eight match points before he grabbed a vital serve and held to win the match.