Arsenal reached the FA Cup semi-finals with a 4-1 win over Everton and Manchester United kept alive its faint top-four hopes in the Premier League by beating West Bromwich Albion 3-0.
Substitute Olivier Giroud scored two of Arsenal's three goals in the final 22 minutes as his team became the first to advance to the last-four of the cup.
Mesut Ozil scored his first goal in three months to give Arsenal a seventh-minute lead, Romelu Lukaku equalised in the 32nd before Mikel Arteta put the hosts back in front with a twice-taken penalty in the 68th.
In the Premier League, Phil Jones opened the scoring for Manchester United in the 34th minute, heading past Ben Foster following Robin Van Persie's free kick.
United doubled its lead after the interval following a fluid move. A neat exchange between Wayne Rooney and Juan Mata saw the latter roll the ball wide for Rafael, and his cross was met by a completely unmarked Rooney in the 65th.
Sheffield march on
Substitute Danny Welbeck added a third in the 82nd minute to move United into sixth place, level on 48 points with Moyes' former club Everton.
Elsewhere, relegation-threatened Cardiff City secured a 3-1 win over fellow strugglers Fulham. Southampton beat Crystal Palace 1-0 while Stoke drew 1-1 at Norwich.
Cardiff moved into 18th, level on points with West Brom in 17th place.
Elsewhere, English third-tier club Sheffield United kept alive their FA Cup dream when two goals in as many minutes gave them a 2-0 victory over Charlton Athletic and a place in the semi-finals.
Second-half strikes from Ryan Flynn and John Brayford caught second-tier Charlton cold at Bramall Lane, as the Blades became the first third-tier team since 2001 to reach the competition's last four.