Suarez had failed to get on the scoresheet in his first five games for the Catalan giants after serving a four-month suspension for biting Italy defender Giorgio Chiellini.
The club record signing struck at the sixth time of asking, though, and it was worth the wait as he produced a sharp turn before applying a clinical finish to set Barca on their way to a 4-0 UEFA Champions League win.
Messi then broke Raul's Champions League goalscoring record by surpassing the Real Madrid legend's tally of 71 and did so in style, by moving on to 74 goals with a hat-trick in Nicosia on Tuesday.
Suarez ought to have had more than one goal as Barca, who were already assured of a place in the last 16, dominated the Cypriot champions, but the Uruguay striker was just happy to get off the mark.
The close-season signing from Liverpool said: "I'm very happy at the moment because I'm at the club I wanted to be at, I'm helping out and the most important thing is that the team is completing its objectives.
"It's always important for a striker to score goals and after going a few games without scoring, I was finally able to do that.
"We knew that APOEL would keep it tight, that it was going to be difficult to find space, but we played a very good first half and it was vital that we opened up their defence."
The Catalan giants head into their final Group F clash against Paris Saint-Germain next month needing a victory to win the pool and Suarez is determined to finish off the job against a side who beat Barca 3-2 at the Parc des Princes.
He said: "We know that Paris are a very strong team, they are going to be very difficult to beat, as we found out the hard way when we played them. But now we have to be strong at home so we can get first place."
APOEL are bottom of the group, but can secure a UEFA Europe League spot by beating Ajax in Amsterdam.
