The visitors were quickest out of the starting blocks, with Henrikh Mkhitaryan picking up a loose ball outside the Saints box and rifling goalwards eight minutes in, but Fraser Forster was able to save.
Marcus Rashford then curled a free kick inches wide from 25 yards out later in the half, but United did not add to their goal tally come the break.
Forster then saved with his feet from Lukaku's low effort, before Marouane Fellaini cleared off the line from Romeu's header as Southampton went up the gears.
Substitute Ander Herrera fired over the Saints' bar while Romeu slid the ball just wide from an angle, with United forced to defend stoutly in injury time to hang on to all three points.
Mourinho was sent off by referee Craig Pawson deep in stoppage time for venturing onto the field of play, the Portuguese coach calmly shaking hands with the Saints coaches before heading down the tunnel.
The result maintains United's unbeaten start to their league campaign, although Manchester City's 5-0 win over Crystal Palace keeps the Red Devils off the top of the table in second place.