There was no requirement in the UK to keep any records of departing passengers. There was requirement to prepare a passenger list for US Immigration. That was often compiled on embarkation but in most cases the only surviving copies were those presented on arrival in the US. However there are some passenger lists in PRONI Belfast (apparently local copies of the US immigration lists, kept by the ships agents in Ireland which have survived). For the period you are interested in in, PRONI have passenger lists for Philadelphia 1800 – 82, Baltimore 1820 -91, New York 1826 – 27, 1840 – 42 & 1850 -52. PRONI also has a personal names index for emigrants, with information gathered from emigrant letters and similar sources.
None of these records are on-line as far as I am aware. You would need to search in person or engage a researcher.
You could also try the Emigration database at the Centre for Migration Studies at the Ulster American Folk Park. http://www.qub.ac.uk/cms/
Be aware that although many emigrants did join ships that sailed directly from Ireland to the US, by far the busiest port of departure was Liverpool (with the ships agents often throwing the cost of the short passage from Ireland to Liverpool in free, as part of the price of the ticket). So your ancestors might have left from there, rather than from Ireland itself.