Manchester City were held to a shock draw at home by Southampton, with the visitors even having a good penalty shout turned down.
Liverpool grabbed a 3-0 win over Crystal Palace, with goals from Sadio Mane, Mohamed Salah and Naby Keita getting the job done against the Eagles at Anfield.
Gillespie insists that Liverpool did not hit the levels they hit against AC Milan in midweek, but feels doing that every game is impossible.
He stressed the importance of finding wins when not playing in top gear and noted Manchester City again dropped points.
Gillespie said post match on LFC TV: “In my day we played 42 [league] matches and you can’t play to that level every week.
“It’s about winning matches, getting the three points.
“Then we find out that Manchester City have slipped up again.
“So it’s a big three points for us.”
Liverpool now sit top of the Premier League table with 13 points from five games, a total which gives them a cushion of three points over Manchester City.
The Reds are next due to travel to Norwich City in the EFL Cup before taking a trip to Brentford in the Premier League next weekend.