Around 32,000 fans from Swindon made the trip to Wembley for Swindon's first game there since in 17 years for the league one play off final.
Neither side started well, with Swindon barely having any chances in the first half.
Millwall had the ball in the net first when Amankwaah looked to have scored an own goal, but this was ruled out for offiside.
Millwall finally took the lead six minutes before half time with a close range goal from Robinson following a Millwall corner to make it 1-0.
Swindon looked a much better side in the second half, and started to look as if they would equalise, but it was not to be.
Swindon's best chance of the game fell to Charlie Austin who only had the 'keeper to beat but the missed the goal.
In the final minutes of stoppage time Swindon pushed everyone forward to try and force extra time including goal keeper David Lucas, but Swindon ran out of time.
Overall Millwall did look the slightly better side so did derserve to win. Swindon will now have 2 1/2 months off before trying again in in League One in August. However Millwall lost in the play off's last year and won this year, so maybe next year for Swindon.