Use 'patch through' when referring to phone calls or radio communication, not face-to-face talks. The operator will patch through your call to the manager.
Pro Tip 2/3
Not Always Instant
'Patch through' can imply a brief wait while the connection is being made. Hang on, I'm patching you through to her line now.
Pro Tip 3/3
Technical Intervention
There's often a technical process or a person involved in making the connection. She patched through my call using the switchboard.