The Mets (18-18) got seven hits on the night, led by two doubles from Mark Vientos and Domingo Martinez.
Shervyen Newton, Vientos and Kenny Hernandez all recorded an RBI for Kingsport.
The Royals (11-26) were able to get three runs in the bottom of the fourth inning. Chris Hudgins had a two-run home run as part of the rally. Five other Royals had a hit in the game.
In the ninth inning, Kingsport was able to get the tying run to third base in L.A. Woodard but could not cash in on the opportunity as Jarred Kelenic and Newton both struck out.
Ted Cillis (1-0) picked up the win for Burlington, pitching 2 1/3 innings.
Cillis, who entered the game with one out in the second, struck out two while giving up one hit and walking one.
Kyle Hinton picked up his third save by pitching a perfect ninth inning, striking out two.
Matt Cleveland (1-1) took the loss for Kingsport. He went five innings, giving up four earned runs on six hits and four walks. He struck out two.
The Mets will continue the road series today with first pitch scheduled for 6 p.m.