Spurs rode their luck throughout much of the first half in Cyprus on Tuesday but took the lead against the run of play in the 39th minute through Harry Kane.
Hugo Lloris was forced into two crucial saves after the interval as APOEL continued to threaten, but Kane added two more goals to make sure of victory.
"It was a tough game. They were very motivated. It was important to get three points. It was a must-win," Pochettino told BT Sport after the final whistle.
"We were fantastic. Today there were some good performances from players who are not regulars and a hat-trick for Harry Kane, it was a fantastic performance.
"I am happy because in some games maybe we dominate and create lot of chances and cannot score. Tonight we were clinical in front of goal."
Tottenham's win sees the North Londoners join Real Madrid with a six-point advantage over APOEL and Borussia Dortmund at the top of Group H.