The English champions travel to Scotstoun in the opening round of group fixtures on 14 October.
After Saturday's 29-13 win over Dragons, Rennie says Warriors must improve if they want a winning start.
"It was a step up," he said. "But we've a long way to go to put in a display that would challenge Saracens."
Glasgow, who host Zebre next, ran in five tries against Dragons to go back to the top of Conference A after their shock defeat to Southern Kings last week.
But the Warriors head coach wanted his team to be "more clinical" and believes their error count will need to come down playing in Europe.
"When you play teams like Saracens they're going to spread and cover field so we need to be able to adjust to that," explained Rennie.
"What they are very good at is getting scrum penalties, putting you in the corner and driving and going an inch at a time.
"They'll ask a lot of questions of us so we're going to have to be really resolute defensively, which we saw a lot of [against Dragons]. It was much better but we need to look after the ball and apply pressure.
"I was really impressed with our captains and our leaders. We had a good talk on the way back from Africa and they drove a lot of stuff this week."