The visitors began the night second bottom of the table but climbed out of the relegation zone thanks to their first win in nine games.
Liverpool gifted Rodrigo an early opener before Mohamed Salah levelled for the Reds.
Jurgen Klopp's men had not tasted a league defeat in front of their own supporters since losing to Crystal Palace in April 2017, although they did lose six in a row during the 2020/21 season when crowds were banished due to the coronavirus pandemic.
But that proud record came to a shocking end when Summerville wriggled free inside the area and fired into the far corner.
A fourth defeat in their opening 12 league games of the season leaves Liverpool 13 points off the top of the table and eight adrift of even the top four.
Liverpool goalkeeper Alisson Becker stumbled as he tried to reach Joe Gomez's backpass, allowing Rodrigo an easy finish into the unguarded net for the Spaniard's sixth goal of the season.
That was the eighth time in 12 league games this season that Liverpool have conceded first.
The lead did not last long as Salah levelled with his seventh goal in five games against Leeds by steering home Andy Robertson's cross on 14 minutes.