Rooney struck in the 35th minute to claim his 39th European goal, one more than Ruud van Nistelrooy.
Rooney struck in the 35th minute to claim his 39th European goal, one more than Ruud van Nistelrooy, and move to within just one goal of Bobby Charlton's all-time club scoring record of 249.
Further goals by Juan Mata, Jesse Lingard and a Brad Jones own goal completed a win that left United a point below Group A leaders Fenerbahce in second place ahead of their final game at Zorya Luhansk.
"I'm very happy. It was a very similar performance to others that we had in the Premier League, but this one with goals and that makes a difference," said United manager Mourinho.
"We needed four points. We know that one defeat we are out, so we had that little bit of pressure on the shoulders, but the players coped very well with it."
United will go into their last game knowing a draw will take them into the last 32, having avenged their 1-0 match-day one defeat at Feyenoord, who slipped to third in the group.
Thursday's win was a personal triumph for Rooney, 31, who had branded the British media "disgraceful" after he was pilloried for supposedly 'gatecrashing' a wedding while on England duty.
Rooney told a BT Sport reporter the furore had been a "big deal out of nothing".
"I didn't set foot in the wedding and it's a shame to have to speak about it on nights like this," he added.
Charlton, now a United director, was watching on at Old Trafford and could now see Rooney equal his record against West Ham United in the Premier League on Sunday.
Mourinho made six changes to the United team held 1-1 by Arsenal last Saturday, the most notable of which saw Henrikh Mkhitaryan make his first start since September 10.
The Armenia playmaker, a 26 million pounds ($32.4 million, 30.7 million euros) signing from Borussia Dortmund, showed no signs of rustiness and was a source of menace throughout.
While United made a strong start, there was a scare midway through the first half when Sergio Romero had to produce a smart double save to thwart Rick Karsdorp and visiting captain Dirk Kuyt.
Feyenoord had struck late to defeat United in the reverse fixture in Rotterdam, but any thoughts of a repeat were banished 10 minutes before half-time as Rooney rolled back the clock in style.
Picking up the ball on the left touchline inside his own half, he beat Karsdorp, slid a pass infield to Zlatan Ibrahimovic and cantered into the box to collect the Swede's through ball and dink a shot over former Liverpool goalkeeper Jones.
There was a question of offside, and Renato Tapia was left on the deck after a coming-together with Rooney on the edge of the box, but German referee Manuel Graefe allowed the goal to stand.
Energised by the goal, United began to turn it on the second half, with Mkhitaryan particularly prominent.
Mata's elegant chip drew a finger-tip save from the backpedalling Jones, before Mkhitaryan exchanged passes with Ibrahimovic and thrashed a left-foot shot into the side-netting.
A second goal seemed a matter of time and so it proved in the 69th minute as Ibrahimovic's deft pass freed Rooney, who cleverly squared the ball for Mata to slide it home.
Jones denied Ibrahimovic after his weak kick had gone straight to Mkhitaryan, but he was more obliging moments later when he inadvertently deflected the Swede's cross into his own goal.
Lingard, a late replacement for Mata, was twice thwarted by Jones, but prevailed at the third attempt with a curling left-foot shot.
"Four-nil is a big defeat," said Feyenoord coach Giovanni van Bronckhorst.
"We could cope with Manchester for 70 minutes, but after 70 minutes you could see the difference in quality."