West Ham refused to lie down and Scott Parker gave them hope with a close-range finish in the 66th minute.
Despite maintaining the pressure they could not get the equaliser and slumped to their first defeat on the road in the Premier League in eight matches.
Megson said: "We seem incapable of going on to make it three or four goals after going 2-0 up.
"It happened against Tottenham and Blackburn and it happened again today. We showed apprehension and started playing differently.
"But it would be wrong to criticise the players. We have won what I felt was a vital game for us - our biggest of the season.
"We got off to a great start but we need to show more confidence in ourselves and a bit more belief. We have got good players."
Taylor made the breakthrough with a stunning free-kick from 20 yards that looped over goalkeeper Robert Green.
West Ham were still catching their breath when Davies added a second two minutes later with his 10th goal of the season after good work from Johan Elmander.
Megson said: "They were both fantastic goals. We are not supposed to do those things. It is meant to be kick and rush.
"You will not see a better free-kick taken throughout the world. I really mean that.
"The second goal was really well worked. It is Kevin's best return in the Premier League and there is still a third of the season to run."
However, Parker's effort which came after a cross from Jonathon Spector set up a rousing finale and Bolton's nerves were stretched to breaking point by the end of the game.
Not only did West Ham lose the points, defender James Collins was stretchered off with a hamstring injury after just 15 minutes.
Manager Gianfranco Zola said: "I do not know how long he is going to be out but it does not look good. He will have a scan tomorrow.
"It is a pity that we will lose him for a while as James has been playing well."
Zola felt his side should have got something from the match.
"My only complaint is we did not take the chances we created, " he said. "The performance deserved something more.
"But this is football. You cannot afford to concede to Premier League teams. It is a good lesson for us.
"We were aware the first 15 minutes would be vital as we knew what kind of team Bolton are. So to lose two goals so early in the game was disappointing.
"However they scored a great first goal. He put it right in the top corner and you cannot do anything about a free-kick like that.
"Perhaps we should have done better for the second goal. That was the difference between the teams. No-one will disagree with me that we played the better football."