Page's team still carried their threat going forward in the second period, Bradshaw and Wright forcing saves from Waller.
They were kept in the game by earlier interventions from Davies, but he was eventually beaten twice by Henderson in the final quarter of an hour.
Rochdale's 41-year-old forward flicked in Ed Francis' cut-back and then lobbed the Liverpool 'keeper.
But as they had done all night, the Reds kept on battling and were given hope in the 94th minute with Trueman's tap-in from Oliver O'Connor delivery.
And the recovery was complete in the sixth minute of added time, with Sonni-Lambie following up on the rebound of a saved Trueman hit.
**Team**
Liverpool: Davies, Chambers, McConnell (Onanuga, 63), Bradshaw, Ndiaye, Danns (Trueman, 83), Wright, O'Connor, Cisse, Lucky, Laffey (Sonni-Lambie, 83).
Unused subs: Hall, Farkas, Cowley, Ayman.
Goals: Wright (42), Trueman (90+4), Sonni-Lambie (90+6).
**Next up**
The U21s' next outing is their second fixture in northern Group H of the Jersey Mike's Trophy – a visit to Doncaster Rovers of League One.
The encounter at Club Doncaster Sports Village will be played on Tuesday October 6 (7pm BST kick-off).