Our molecular simulations reveal that wild-type influenza fusion peptides are able to stabilize a highly fusogenic pre-fusion structure, i.e. a peptide bundle formed by four or more trans-membrane arranged fusion peptides. We rationalize that the lipid rim around such bundle has a non-vanishing rim energy (line-tension), which is essential to (i) stabilize the initial contact point between the fusing bilayers, i.e. the stalk, and (ii) drive its subsequent evolution. Such line-tension controlled fusion event does not proceed along the hypothesized standard stalk-hemifusion pathway. In modeled influenza fusion, single point mutations in the influenza fusion peptide either completely inhibit fusion (mutants G1V and W14A) or, intriguingly, specifically arrest fusion at a hemifusion state (mutant G1S). Our simulations demonstrate that, within a line-tension controlled fusion mechanism, these known point mutations either completely inhibit fusion by impairing the peptide's ability to stabilize the required peptide bundle (G1V and W14A) or stabilize a persistent bundle that leads to a kinetically trapped hemifusion state (G1S). In addition, our results further suggest that the recently discovered leaky fusion mutant G13A, which is known to facilitate a pronounced leakage of the target membrane prior to lipid mixing, reduces the membrane integrity by forming a 'super' bundle. Our simulations offer a new interpretation for a number of experimentally observed features of the fusion reaction mediated by the prototypical fusion protein, influenza hemagglutinin, and might bring new insights into mechanisms of other viral fusion reactions.