Van Persie notched on 16 minutes, Cleverley scored three minutes later and Rooney added a second-half effort before substitute Fraizer Campbell headed a consolation against his old club for the Black Cats.
The 3-1 win came after Manchester City had beaten Newcastle by the same scoreline in the early kick-off and handed the advantage back to United in the title race.
Buoyed by their last-gasp 3-2 derby win over City on Sunday, Sir Alex Ferguson’s side started strongly at Old Trafford, with Phil Jones forcing Simon Mignolet into an early save.
The Sunderland keeper then had to tip over a powerful dive from the edge of the box from Ashley Young, while Patrice Evra shot over soon afterwards.
United’s pressure paid off when Van Persie slotted home from eight yards out after John O’Shea deflected Young’s cross into his path.
Mignolet then produced a great save to his left to deny a Van Persie header, but could do nothing to stop Cleverley’s shot into the bottom corner after the United midfielder had played a delightful one-two with Michael Carrick to open the Sunderland defence up.
Now well on top, Rooney shot wide after Van Persie had teed up his strike partner, and then headed just wide from Evra’s cross.
But Sunderland refused to lie down and accept defeat. First James McClean forced David De Gea into a near post save and Evra had to head out Stephane Sessegnon’s powerful effort.
McClean’s long-range effort then took a slight deflection off Cleverley but De Gea saved as United went into the interval with their two-goal lead intact.
Soon after the restart, Titus Bramble deflected Rooney’s close-range effort onto the bar, while at the other end, De Gea produced a great double save to first deny Craig Gardner and then Sessegnon.
Van Persie shot wide while trying to chip the onrushing Mignolet, before he did well to tee up Rooney for United’s third on 59 minutes to seal the three points.
United club captain Nemanja Vidic came on for his first appearance since September but could not stop substitute Campbell heading home Sessegnon’s cross from close range as Sunderland earned a consolation goal with 18 minutes remaining.