Allardyce was pleased with what he saw.
"We are showing how much we are improving," he told BBC Radio 5 Live.
"That was one of the best 10-man performances I have ever seen in my time.
"I have actually won a lot of games with 10 men, and this one well deserved a win with 10 men, but unfortunately we have to kick ourselves for missing golden opportunities."
The Albion manager added: "It makes it more difficult because one more game has ticked by and we have not gained any points on teams near by to put them under pressure and make them think we are on our way and catching up.
"We are very, very close but unfortunately - because of our finishing today and against Manchester United - not close enough."