The game was hastily rearranged after the match at the weekend was abandoned due to a bomb scare and the first half was very low key in front of a sparse Old Trafford crowd.
The one moment of quality produced a fine goal, however, just before the interval when United's French forward Anthony Martial injected pace down the left and Rashford stepped over his cross to leave Rooney with a simple finish.
The 18-year-old Rashford provided another example of his burgeoning talent with a clinical finish, his eighth goal since making his debut in February, in the 74th minute and Ashley Young added United's third before defender Chris Smalling's stoppage-time own goal gave Bournemouth a consolation.
Meanwhile Hull City will face Sheffield Wednesday in the Championship playoff final after surviving a huge scare in a 2-0 home loss to Derby County in their semi-final second leg which left them with a 3-2 aggregate win.
Steve Bruce's Hull side conceded two first-half goals but hung on to secure a place in the final at Wembley at the end of the month, which will bring a money-spinning spot in the Premier League worth more than 300 million dollars for the winners.
Hull finished fourth in the table, five points ahead of fifth-placed Derby, and will play in an all-Yorkshire final against a Wednesday side who completed a 3-1 aggregate victory over Brighton & Hove Albion yesterday.