Minutes after their introduction, the Greek left-back had the option of playing it down the line for Milner - but declined and instead came inside with a ball to Takumi Minamino.
The Yorkshireman was quick to voice his displeasure, as the touchline mic picked up his cry of: "Ah Kostas! F*****g hell."
Reacting to the victory afterwards, Milner posted on Instagram: "Wet, boggy... and got caught warmly welcoming Kostas to the team #welcometoliverpool".
Tsimikas replied with "Thanks bro" along with a laughing emoji and a red heart.