Please see my response to a forum post from a few years ago:
https://my.originlab.com/forum/topic.asp?topic_id=19889
If I apply your data to fit it with WolframAlpha method I mention in that post, I get identical results to the results you got in Origin.
exponential fit {0.1119,6.16622},{0.1939,6.83858},{0.3022,4.9636},{0.4974,3.65903},{0.761,4.25756},{1.0045,3.54405},{2.004,1.82635},{2.9988,1.53728},{3.9991,0.40259},{5.0028,0.1819}
Thanks,
Chris Drozdowski
Originlab Technical Support