His defeat, which saw him hit an uncharacteristic 14 double faults and commit 49 unforced errors, means that for the first time since 2017 he will end the season without a Grand Slam title.
Bagnaia completed the sprint-race double at the RedBullRing circuit to move ahead in the championship standings, with Spaniard Jorge Martin coming in second on the day.
Belarusian Sabalenka posted a 6-3, 6-3 win over World Number 1 Swiatek in the semifinal of the event in Cincinnati on Sunday in a hard fought contest.
Chhetri's goal from the penalty spot in the 51st minute took his tally to 64 goals, overtaking previous all-time leading scorer Bartholomew Ogbeche's total of 63 goals.
Sunday's match at Atletico's Estadio Metropolitano was halted for 15 minutes after home fans threw objects including lighters at Madrid players.