Brown's 23rd-minute spot-kick was saved by Milton Keynes goalkeeper Adolfo Baines, who produced a string of impressive saves to deny ex-Dons boss Wilson his first away win with Pools.
Hartlepool dominated the early exchanges, but Scott Taylor missed a great chance to put the Dons ahead on the break in the tenth minute when he raced onto Gareth Edds' ball over the top.
He headed it past on-rushing Pools' goalkeeper Dimitrios Konstantopoulos but his shot from a difficult angle hit the side-netting.
Jon Daly squandered Pools' first clear chance two minutes later when he stole in at the back post to head Anthony Sweeney's pin-point cross badly wide from six yards.
Jon-Paul McGovern's 30-yard curler was grasped at full-stretch by Konstantopoulos, before Brown broke the offside trap at the other end, only to send an angled drive from 12 yards straight at Baines.
McGovern went close again for the Dons when his stinging angled drive from Aaron Wilbraham's lay-off was saved well by Konstantopoulos at his near post.
Drissa Diallo's clumsy challenge on Matty Robson in the box then gifted the visitors a penalty. Brown's well-placed spot-kick was saved at full stretch by Baines, who reacted quickly to deny Ali Gibb from the rebound.
Hartlepool started the second half brightly and Brown's clever flick gave the unmarked Robson a great chance to put the visitors ahead three minutes after the re-start, but his angled drive from 15 yards flew just wide of the far post.
Gary Liddle then saw his stinging 30-yard strike fly inches wide of the post with Dons keeper Baines rooted to the spot, before the unmarked Sweeney somehow turned Robson's low cross over the bar from four yards.
The Dons finally settled into their stride in the last half hour and Clive Platt's volley from Lloyd Dyer's corner was cleared off the line by Pools defender Liddle in the 63rd minute.
McGovern's 25-yard free-kick flew just over the bar in the 78th minute, before substitute Eifion Williams' close-range volley from Robson's cross was well saved by Baines in the 85th minute.
Platt almost won it for the Dons three minutes from full-time but his close-range toe poke came back off the bar and he headed the rebound over.