Beating Liverpool was a huge result for Moyes, who had come under fire after presiding over United's worst start to a league campaign in nine years in his first season in charge after taking over from Alex Ferguson. Only three days ago, the champions had been thrashed 4-1 by neighbor Manchester City.
"I think what was important (was) that the whole club showed we were hurt by the game on Sunday, and the players have gone out and done a great job," Moyes said.
"I've said I will fix it and turn it around, and I will do."
With no European football this season, Liverpool is left with just two trophies to play for and will rue that its full-strength lineup missed a great chance to beat a weakened United for a second time this season, having already won 1-0 in the league on Sept. 1.
An already high-profile match was given a further edge by the comeback of Suarez, who played a World Cup qualifier for Uruguay as recently as two weeks ago but hadn't worn a Liverpool shirt for a competitive match since April 21 when he sunk his teeth into Ivanovic's upper arm.
In those five months, his relationship with the club had been severely tested after he accused Liverpool's hierarchy of breaking an agreement to sell him if it failed to qualify for the Champions League. Liverpool turned down two bids from Arsenal for Suarez over the summer.
He was nevertheless well-received by Liverpool's travelling support at Old Trafford, although his reception from the home fans was typically hostile having been banned for eight matches in 2011 for repeatedly racially abusing United defender Patrice Evra during a league match.
With Liverpool changing formation to a 3-5-2, he played up front with Daniel Sturridge, with the pair initially posing problems on the counter-attack. Suarez miscontrolled when he could have run through on goal and Sturridge toe-poked wide after getting half a yard on his marker.
United was poor in possession in the first half but when Jose Enrique lost Hernandez at a corner soon after the break, the Mexico striker leapt to guide home an impudent volley from inside the six-yard box, provoking an air-punching celebration from Moyes.
Suarez shot into the side netting in the 70th, then sent a free kick crashing against the bar two minutes later as United held off a strong finish by Liverpool, a record eight-time winner of the competition.
"It was a massive turnaround (from the weekend), especially with players who haven't played a lot of games this season," United defender Chris Smalling said. "There was a lot of doubt after the weekend so we had to go out and get a performance and we did."
Swansea conceded three times in 24 second-half minutes at St. Andrew's to exit the competition it won last season for a first major title in the club's history. Wilfried Bony grabbed a late consolation for the Welsh club.
Nacho Monreal scored the winning penalty for a weakened Arsenal team at The Hawthorns after Craig Dawson and Morgan Amalfitano missed the target from 12 yards for West Brom earlier in the shootout.
Thomas Eisfeld put Arsenal ahead in regulation time, scoring in the 61st minute after running onto a pass from striker Nicklas Bendtner, who was making his first start for Arsenal since August 2011. Saido Berahino sent the game into extra time by equalizing 10 minutes later.
Papiss Cisse scored his first goal since April in Newcastle's 2-0 home win over Leeds, while Stephen Ireland and Peter Crouch were on target for Stoke in a 2-0 victory at Tranmere.
