United have now scored 16 goals in their last five league games to close the gap on Chelsea and Leicester in the top four race.
That firepower was in evidence as the game was turned on its head within two minutes as Martial showed great strength and touch to tee up Rashford to slot home his 21st goal of the season.
Both Rashford and Martial are enjoying their best ever scoring seasons and moments later it was the Frenchman's turn to score his 21st of the campaign as he powered past Kyle Walker-Peters before lashing a shot past Alex McCarthy for his 50th Premier League goal.
Southampton were lucky to get into half-time with just 11 men on the field as Oriol Romeu somehow escaped sanction despite a VAR review for a dangerous lunge on Mason Greenwood.
The Saints stemmed United's flow at the start of the second half and Solskjaer responded by replacing Pogba with Fred.
Rashford and Martial still had chances to kill the game as Ryan Bertrand's block denied the English international a certain second before Martial slashed a shot wide after a purposeful run from the halfway line.
United were given a warning five minutes from time when De Gea produced a stunning save to turn Redmond's effort over to bounce back from a couple of errors leading to goals since the Premier League's restart.
The hosts were left down to 10 men for the final few minutes as Brandon Williams was taken off with a head injury after Solskjaer had used his allocated three stoppages to make substitutions.
And they were caught out as Obafemi was in the right place to claim a point for Southampton.