Nolan turned a low cross from Ricardo Vaz Te, who was initially flagged off-side only for the assistant referee Simon Long to change his mind.
Aston Villa players complained but referee Mike Dean confirmed the goal and it was enough to give the Hammers a winning start on their return to the Premier League.
Manager Sam Allardyce wrote in his match-day notes how he wanted to maintain the "feel-good factor" that was generated in east London by the Olympics.
Nicola Adams, the first women's boxing Olympic champion, was among a number of Team GB members invited to the match by West Ham, who are anxious to prove they can build on the London 2012 legacy.
No decision has yet been made on who will take over the tenancy of the Olympic Stadium but club co-owner David Gold described West Ham as the best option in his programme notes.
Whatever the arguments for and against West Ham moving, there is no debating the fact that the 34,172 who were at Upton Park today would rattle around inside the Olympic Stadium.
Adams, Mark Hunter, Jo Pavey, Andy Baddeley and Yamile Aldama were given a rousing reception when they were introduced to the crowd, with West Ham flags draped around their shoulders to show the Olympic spirit is still resonating around the east end.
But unfortunately there was nothing as classy as the Ali shuffle, which Adams produced in her historic Olympic final triumph, on display at Upton Park.
West Ham featured two debutants in goalkeeper Jussi Jaaskelainen and midfielder Mohamed Diame plus the returning James Collins, who lined up against his old team.
With Paul Lambert in charge for the first time, Aston Villa fielded four summer signings, including Brett Holman whose early 20-yard drive briefly had Jaaskelainen worried.
Holman was involved again after receiving a reverse pass from Stephen Ireland but his cross was cleared by Winston Reid, one of three timely interventions from West Ham's Kiwi defender.
West Ham did not create any real pressure until Matt Taylor played Carlton Cole into the box, forcing Villa to concede a 14th minute corner. But nothing came of it.
Villa looked marginally the more dangerous going forward - in relative terms - but they were denied by a double block from the dominant Collins, who charged down Karim El Ahmadi and then Ireland.
The game began to spark into some sort of life after 36 minutes, when Cole was brought down on the edge of the Villa box by Ron Vlaar.
Mark Noble touched the free-kick to Nolan, who drilled it low and through the wall but just wide of Shay Given's right-hand post.
Four minutes later, West Ham had the lead. Noble's lofted free-kick fell to Ricardo Vaz Te, who pulled the ball back for Nolan to score from close range.
But there was more than a hint of controversy about it. Assistant referee Simon Long initially flagged for off-side, in the belief that Collins had flicked the ball on to Vaz Te before changing his mind.
Villa players complained immediately as West Ham celebrated, and referee Mike Dean consulted with his assistant before confirming the goal.
West Ham began to grow as the game went on and Cole was convinced he should have had a penalty when he went down under a challenge from Vlaar.
The Hammers could have doubled their lead 65 minutes, when Vaz Te skipped brilliantly past Nathan Baker and cut a low cross back to Nolan, whose shot was blocked by Vlaar and cleared.
Just after the hour mark, Matt Taylor swung in a dangerous free-kick from the right touchline which Cole rose to meet but the former England striker directed his header wide.
Vlaar tried his luck from 35 yards out but the final word could have gone to West Ham replacement Mo Maiga on his debut.
The Mali striker, on for Cole, was released by a long ball as Villa chased the equaliser and he rounded Given but Baker got back to clear off the line.