Rashford scored twice against Charlton as United won the Carabao Cup quarter-final 3-0 at Old Trafford.
The 25-year-old has now found the back of the net in his last eight home matches in all competitions.
Rooney tweeted, “Class @MarcusRashford. Let’s keep it going” alongside several applause emojis.
Rashford’s goal against Everton in the FA Cup was his seventh successive home fixture, equalling a record Rooney set back in 2012.
The Manchester Derby on Saturday will now present Rashford with the chance to equal Dennis Viollet’s club record of scoring in nine consecutive home games.
Rashford’s impressive run began in the Europa League against FC Sheriff and he already has 15 goals this campaign.