City were 2-0 up inside 15 minutes, Nigel de Jong powering home his first goal for the club and then Pablo Zabaleta claiming a second via Lars Jacobsen's attempted clearance.
The Hammers hit back in the 33rd minute through Demba Ba, but it was the hosts who looked more likely to score another after the break and deservedly made off with all three points which sees them pull seven clear of their nearest rivals in the battle to claim the fourth and final Champions League berth.
"It was a bit nervy but I think if we'd needed to step it up another gear, we would have done," City goalkeeper Hart told Sky Sports. "We got the two goals early and I think it's fair to say got sloppy and let them back in it, but I think if needs be we'd have got another goal."
De Jong's opener was something of a surprise, the midfielder not famed for his goalscoring prowess. The Dutchman's afternoon turned sour at the break however, when an injury meant he was unable to come out for the second half, but Hart felt De Jong had nevertheless made his mark.
"The whole stadium's in shock and I think it still is," he joked. "I think that's why he had to come off actually.
"He doesn't usually do that but it was a great moment for him."
City defender Vincent Kompany, meanwhile, felt the result was indicative of the strides taken under manager Roberto Mancini this season.
"Maybe last season we would have lost points in a game like this, we're really happy because it wasn't easy," he said.
Asked about the increasing likelihood of sealing their place in the Champions League next season, and perhaps even closing the five-point gap to third-placed Arsenal, the Belgian added: "It's massive, not just for the group but for the club.
"We really want to achieve this and, why not catch up with Arsenal? We're going to try it."
Mancini, meanwhile, was keen to clear up the confusion over who should be credited with City's second goal, after Zabaleta saw his goal-bound shot deflected in off the bar by Jacobsen.
"It was Zabaleta," said the Italian. "Zabaleta did everything - a one-two, the shot on goal. It was Zabaleta's."
Mancini admits his players may have been guilty of thinking their job was done after they took an early two-goal lead.
"At this moment, at the end of the season, every game is difficult and if you play against a team that is trying to avoid relegation, it is very, very hard," Mancini said.
"I think we started the game very well, playing fantastic football and scoring two goals.
"Probably after these two goals we thought it was finished, but it never is until the referee blows the final whistle."
Mancini feels his team deserve to be where they are but wants to see no let-up from them until the season has come to an end.
"We can only relax after May 22, not before," said Mancini, who also confirmed that De Jong had come off at half-time merely as a precaution with a view to next weekend's clash with Everton.
"We should try to get the maximum number of points that we can.
"Everything is in our hands because for all the season we have stayed at the top between first and fourth position.
"I think after what we have done, we deserve to be in the Champions League, but we have another four games - it is not finished.
"We are not under pressure, but we know we have a big chance.
"Our target at the start of the season was the Champions League and now, we are in a good way.
"We are in fourth position and have seven points over Tottenham and Liverpool, and we will play the FA Cup final (against Stoke on May 14).
"I think we have done a good job up to now."
Although the loss was West Ham's fifth in a row and they remain bottom of the table and three points from safety, manager Avram Grant is staying upbeat.
"We made a very bad start - in the first 20 minutes we were not in the game, they had scored two goals and could have scored more," Grant said.
"We had a great comeback and showed a lot of spirit and the character of the team, and we made it 2-1 - also Robbie Keane had a big chance.
"In the second half we could have done some more, so there was a lot of positives to take for the next games, which are the most important ones."
The Hammers play Blackburn, Wigan and Sunderland in their final three games and Grant has set a target of at least seven points from those fixtures to beat the drop.
"Now we play against a team that are in our area (Blackburn), so we have to win this game," Grant said.
"I thought that other teams would take more points in the last two games, so the gap stays three points.
"Our target is to take seven points - and we will try to take all nine - but we are playing against teams that also want the points.
"But we believe if we use the character we have shown in past games away against top teams, we can win games."
