Cole, who spent last season on loan at Lille, lasted just 23 minutes before he was replaced by 17-year-old Raheem Sterling at the Central Stadion in Belarus.
"Joe just felt his hamstring a little bit," Rodgers said. "I don't think it is totally pulled and it was just a little bit of tightness so we had to take him off.
"It disrupted our plans a little bit but it was better to have him come off rather than risk injury."
"Glen was only scheduled to play 45 minutes anyway so he was taken off as planned," he added.
Stewart Downing's second half strike ensured Rodgers' first competitive match in charge ended in victory, and the Ulsterman was happy with what he saw at the Central Stadion in Belarus.
"It will take time until Liverpool start playing the way Swansea did last season (tactically)," he said. "Actually the way you play depends on the players you have. As for today I saw two things which were very important for me, I saw desire and commitment. I am very grateful to my players for a good game."
He added: "Obviously it was a serious test of the players' fitness and I am really happy with the result. This was my first competitive game in charge of Liverpool and for me it didn't really matter whether the game was played in Belarus or Brazil as I am really honoured to manage this side."
However his first match at the helm did not go all Liverpool's way. The hosts, who are halfway through their domestic campaign, dominated the majority of the game and were it not for poor finishing would have emerged victorious.
Rodgers continued: "This game was a great test for us at this point of the season. Today we saw that Gomel is a very good club which is already into the season.
"Their tactical idea was to defend and counter-attack so it was difficult for us to find the open spaces in the first half but at half-time I told my players to be patient and that the chances would come.
"After the first 45 minutes I also changed the structure of the midfield and Steven Gerrard started to play a bit lower than the other two central midfielders. I am glad that Stewart scored the goal as we put him on that side to come in and shoot. It was a great shot."