Boro's Cristhian Stuani struck twice in the first half to earn the points, while a Michail Antonio header gave West Ham the spoils against a Bournemouth side reduced to 10 men in the 78th minute when midfielder Harry Arter was sent off.
Both matches were laced with added importance as Boro and Sunderland renewed their rivalry having last faced each other in the league nearly eight years ago and, with Newcastle United relegated, the clash is the northeast's only derby this season.
For West Ham manager Slaven Bilic anything less than victory in the club's first top flight game at their new ground would have raised questions "like a snowball" about the wisdom of the move from Upton Park to their new 60,000 capacity stadium.
Sunderland, who lost their league opener at Manchester City, had five players making their home debuts so it was a case of bedding in the new faces but Moyes' young side really struggled to contain Middlesbrough's accomplished attack.
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Boro's Uruguay striker Stuani was the difference, first with some individual brilliance after 12 minutes and then by rounding off a superb team effort before the interval. Dutchman Patrick van Aanholt pulled one back for the hosts but it was too late.
"It's always important to win, especially in a derby and for our crowd who have been waiting for this moment for a long time, so I'm pleased for the players and the crowd," said Boro manager Aitor Karanka, whose side are sixth with four points.
LONDON CHEER
In London, it was four years ago that the Olympic Stadium hosted the Games but on Sunday, as Rio prepared to bring the curtain down on the latest edition, there were again plenty of flags as West Ham sought to give their fans something to cheer.
The home supporters endured a nervous wait as Bilic's side, missing forwards Andre Ayew, Andy Carroll and Dimitri Payet through injury, struggled until Bournemouth had influential midfielder Arter dismissed for a second bookable offence.
That opened up space at the back and seven minutes later Antonio rose at the far post to head home a beautifully flighted cross from Gokhan Tore. The win moved West Ham, who lost their opening game at Chelsea, up to 10th on three points.
"It was not a good game but the lads were brilliant," said Bilic. "It was good in terms of effort, team spirit, quality of aggression. It's important to win because people would start to ask questions. Is it a proper home? It'd be like a snowball."
Bilic said Carroll, who has a long injury history, will be out for up to six weeks with a thigh problem. The Croat added that the club would try to bring in at least one player before the transfer window shuts at the end of the month.
"I will speak to the chairman tomorrow. If we get one, hopefully it will be a good one," he said.
The defeat left Bournemouth at the bottom of the table without a point behind Crystal Palace, who lost at Tottneham hotspur on Saturday, and Sunderland on goal difference.
(Reporting by Neil Robinson; editing by Ken Ferris)
