Warriors lost 38-19 to the English champions, who topped Pool Three for a home quarter-final as top seeds.
Through as one of the best runners-up, Glasgow are likely to return to Allianz Park in the last eight, where they were beaten 38-13 at the same stage in 2017.
"We know we'll be better next time - we'll need to be," Rennie said.
"We're going to have to defend better and tackle better as individuals and be a bit more clinical - turn some of that second-half pressure into points."
The lead changed four times in a frantic first half as each side scored three tries, Glasgow going in at the break 24-19 down.
Jonny Gray failed to reappear for the second half and the Warriors also lost fellow Scotland Six Nations squad members Ryan Wilson, Sam Johnson and Grant Stewart to injury in the second period.
Saracens replacement lock Will Skelton scored one try and set up another for Maro Itoje as the hosts' power saw them stretch clear in the final quarter, despite Glasgow's best efforts.
"We gave them a lot of ball to hurt us," Rennie added. "At various times they looked pretty shattered but the only way you can run them around is by maintaining ball and pressure.
"I thought maybe we kicked a little bit too much ball away aimlessly in that last part of the game. But whoever we play (in the quarter-finals) or wherever it is, we'll give it plenty."
Rennie confirmed forwards Wilson, Gray and Stewart all have shoulder injuries, while centre Johnson failed a head injury assessment for concussion.
"They'll all be screened over the next few days to see the severity - hopefully it's not too bad," added the New Zealander, who was unable to confirm a timescale for any of the four.