Ex-Celtic striker Rasmussen was given the nickname 'Duncan' during his early years with AGF Aarhus as he reminded a coach of former Everton forward Duncan Ferguson with the Denmark international even sporting the moniker on the back of his shirt.
And he provided the moment of inspiration to send the Danish champion into the Europa League group stages, with Ronald Koeman's Southampton losing 1-0 on the night and falling at the final hurdle of qualification.
Having made heavy weather of a first-leg draw at St Mary's a week ago, Southampton was unable to get on the scoresheet at the soaked MCH Arena in Herning and will now go back to focusing on the stuttering start it has made to its domestic campaign.
After passing up several chances to open the scoring and chalk up a crucial away goal, Jay Rodriguez was guilty of a sloppy pass on the edge of his own penalty area that led to Rasmussen slotting home the decider.
Saints were nearly ahead inside opening four minutes as captain Fonte threw himself at Ward-Prowse's corner only to see his goalbound header cleared off the line by Jakob Poulsen.
The visitor was in behind again moments later with Rodriguez opting to pass rather than shoot, leading to a Ward-Prowse effort cannoning away off the covering defender.
With heavy rain setting in, Southampton should have been in-front at the midway point of the first-half as Davis turned in the box before shooting low at Midtjylland goalkeeper Johan Dahlin, who pushed the effort away and into the path of the offside Rodriguez.
And it was Rasmussen who showed Saints how to finish, latching onto Bach Bak's pass after Rodriguez had gifted possession to the Midtjylland skipper and tucking away his first chance of the night with aplomb.
Pione Sisto could have made things worse for the English Premier League side but he lacked the composure to beat Maarten Stekelenburg when inside the box.
Rodriguez, looking to atone for the role he played in Southampton falling behind, rose well to meet Cuco Martina's cross but his header was too close to Dahlin to level.
After the break it was Rasmussen left kicking himself as he could only head Jesper Lauridsen's free-kick wide of Stekelenburg's post despite poor marking by the Saints' backline.
With Southampton pushing men forward in search of an equaliser, Midtjylland was starting to find more space on the counter-attack and referee Liran Liany had a big decision to make when Matt Targett brought down the marauding Rilwan Hassan - with a yellow card shown to the young full-back.
Daniel Royer should have sewn up the tie with 10 minutes remaining but his rising effort from the edge of the area flew inches over Stekelenburg's crossbar but it did not prove costly as a strong rearguard held off late Southampton pressure to end its brief European adventure.
