Suarez marked his return to Anfield since April with a first-half opening goal for coach Brendan Rodgers’ side against Crystal Palace.
Daniel Sturridge and Steven Gerarrd got Liverpool’s second and third goal while Dwight Gayle netted a consolation goal for ‘The Eagles’.
After a good work by Victor Moses, Suarez passed to the overlapping Jose Enrique and then collected his return ball to fire in from 12 yards despite falling over.
Ian Holloway’s men were two-goal down withing three minutes of conceding when Jose Enrique’s long pass was headed onto Sturridge, who turned Damien Delaney before finishing into the bottom corner.
Palace went further behind when Dean Moxey tugged at Raheem Sterlin’s arm and, although contact was on the edge of the area, referee Anthony Taylor signaled a spot kick after advice from his assistant.
Gerarrd sent Julian Speroni the wrong way to score for the 15th consecutive league season, a club record.
Gayle grabbed a consolatory goal in the 76th minute, heading in Campana’s freee-kick.
The loss was Crystal Palace’s fourth in succession, and leaves them two points ahead of bottom placed Sunderland who are at home to champions Manchester United.
Liverpool are one point clear at the top of the Premier League table, having played one more than Arsenal, which plays West Brom on Sunday.