Horvath was born and raised in Highlands Ranch, Colorado. His father, Peter, played college soccer with the Midwestern State Mustangs and professionally with Major Indoor Soccer League club Denver Avalanche, and his mother, Deana, played high school soccer. Horvath himself grew up playing with U.S. Soccer Development Academy side Real Colorado and for Arapahoe High School. He also trained with Bristol City, Manchester City, and Stoke City in England.
Horvath signed his first professional contract in 2013 with Tippeligaen club Molde in Norway, having been recruited by Molde manager and former Manchester United player Ole Gunnar Solskjær. Horvath first appeared in a Molde matchday squad on August 10, 2013. He made his professional debut on May 16, 2015, replacing Ørjan Nyland in the 55th minute of the match. On July 1, 2015, Nyland was sold to Bundesliga club Ingolstadt 04, leaving Horvath as Molde's starting goalkeeper at age 20. By the conclusion of the 2015 Tippeligaen, Horvath had helped Molde concede just 31 goals for the season, the second-best mark in the league only behind champions Rosenborg.
Horvath backstopped Molde to a 5–1 aggregate victory over Pyunik in the second qualifying round of the 2015–16 UEFA Champions League, before a 4–4 aggregate loss on the away goals rule to Dinamo Zagreb in the third qualifying round. Later, in the playoff round of the 2015–16 UEFA Europa League, Horvath and Molde had a 3–3 aggregate win on away goals over Standard Liège. With Molde thus qualifying for Europa League Group A, Horvath joined Brad Friedel, Kasey Keller, Tim Howard, and Brad Guzan as the only American goalkeepers ever to play in the UEFA Europa League/UEFA Cup group stage.
On October 2, 2015, Horvath was named to the Europa League Team of the Week, a day after being named Man of the Match for a 1–1 draw against Ajax in which he notched 10 saves, most notably on a point-blank header from Viktor Fischer and on a corner serve from Davy Klaassen in stoppage time. Horvath led Molde to a first-place finish in Europa League Group A with successful two-legged fixtures against Ajax, Celtic, and Fenerbahçe, the latter of which featured two saves on Robin van Persie and three saves on Nani. Combining his play in the Tippeligaen and in the Europa League, he had not allowed multiple goals in a match for over two months. In February 2016, Horvath faced eventual Europa League champions Sevilla in the knockout phase and was unable to prevent a 1–3 aggregate defeat of Molde, although he kept a clean sheet in the second leg.
On January 3, 2017, Horvath signed with Jupiler Pro League defending champions Club Brugge on a four-and-a-half year contract. He debuted as Club Brugge's starting goalkeeper on May 5 in the Championship playoffs. After initially letting in a goal due to confusing a whistle from the stands as the referee's whistle, Horvath rebounded to make three saves, and even notched an assist to José Izquierdo off a punt, in a 3–1 victory over Charleroi. The following match, in a 1–1 draw against Anderlecht, Horvath made four saves, including on a penalty kick attempted by Youri Tielemans. Horvath continued to start in goal to close out the season, helping Club Brugge secure a place in the 2017-18 UEFA Champions League third qualifying round.
Horvath has represented United States youth teams at every level since U-14. He was named to the U.S. under-20 squad for the 2015 FIFA U-20 World Cup in New Zealand, although his club did not release him. He was the U.S. under-23 starting goalkeeper for the CONCACAF–CONMEBOL Olympic qualifying playoff against Colombia, conceding two goals in two matches as the U.S. lost on aggregate.
Horvath was named to the 23-man United States men's national team for the Copa América Centenario held in June 2016. Head coach Jürgen Klinsmann called Horvath one of his "biggest hopes for the future” at the goalkeeper position. On October 7, 2016, Horvath made his USMNT debut, playing the full match and keeping a clean sheet in a 2–0 victory over Cuba.As of match played September 16, 2017