Liverpool's cushion was reduced to two on the stroke of half-time, Valentin Joseph converting from a free-kick launched into the box.
It was back to three, though, soon after the restart.
After Bradshaw did well to win back possession in the Brighton third, Wright spun past Callum Mackley and calmly side-footed the ball past Steven Hall.
Wright was denied his hat-trick by a reaction save from the 'keeper, but the resulting Ayman corner was inadvertently flicked in by Theo Outen.
At the other end, Bailey Hall produced a stunning intervention to stop Kofi Shaw handing the Seagulls a lifeline.
**Team**
Liverpool: Hall, Mabaya (Onanuga, 5), Ayman, Cisse, Ndiaye, Upton (Trueman, 88), O'Connor, Laffey (Lucky, 88), W. Wright, Sonni-Lambie, Bradshaw (Danns, 56).
Unused subs: M. Wright, P. Farkas, Cowley.
Goals: Sonni-Lambie (2), Laffey (11), Wright (25, 50), own goal (66).
**Next up**
Liverpool begin their EFL Trophy campaign by travelling to League Two side Rochdale on Tuesday night.
The action at Crown Oil Arena is scheduled to get under way at 7pm BST.