Kansas City Sporting (9-3-6) has won a franchise-record eight straight home games, including all competitions.
"We had a slow start and they capitalized on our mistakes on their two goals," said Montreal midfielder Dilly Duka. "We need to be a better away team and play like an away team."
Feilhaber opened the scoring in the fourth minute after Roger Espinoza's interception in the Impact backfield. Espinoza fed Kristian Nemeth, who crossed it back to Feilhaber for a left-footed blast into the right corner of the net.
Feilhaber then assisted Dom Dwyer in the 34th minute, splitting the Impact's central defence with a 50-yard pass from behind midfield. Dwyer chased it to the top of the 18-yard box where he curled it home.
"The first two mental mistakes cost us the game," said Montreal head coach Frank Klopas. "We pushed in the second half. If you look at the general performance of the team, we played a good game."
Ignacio Piatti scored in the 59th minute for the Impact (6-8-3). The goal snapped Sporting KC's 355-minute MLS shutout streak.