The hosts lost both openers cheaply, as Nick Gubbins was caught off the bowling of James Tomlinson, and Sam Robson edged Tino Best to Jimmy Adams.
But Voges and Malan's 279-run stand put Middlesex firmly on top, before Malan edged Liam Dawson to slip for 147.
Voges remained unbeaten at the close on 128 alongside John Simpson (28) as Middlesex finished the day on 342-3.
Middlesex's third-wicket partnership leaves them in strong position to claim their first win in the County Championship this season, having drawn all six of their games so far.
Voges' ton, his first of the season, came off 200 balls, taking the former Australia batsman beyond 13,000 first-class runs in his career.
Malan was more aggressive in dismantling Hampshire's bowling attack, hitting 17 fours and two sixes before getting dismissed attempting to drive a Dawson delivery.
The visitors could take few positives from their performance, but they will remain hopeful of salvaging a draw from the game, with poor weather forecast for Tuesday's third day.

The family had asked that half the remains be returned to them with the other half scattered in the grounds.
Half were returned but the remainder had been wrongly used in an un-witnessed scattering earlier that day at  Weston Crematorium, Worle.
The two workers lost their jobs and Dignity, which runs the crematorium, apologised for the family's distress.
Instead of admitting their mistake, two members of staff, who have been described as "very experienced", tried to cover it up by using another person's ashes for the ceremony in front of the family.
Eventually, they admitted what they had done to their manager, and Dignity - which runs the crematorium on behalf of North Somerset Council - started an "extensive" disciplinary hearing.
In a statement Dignity said it was an "isolated incident" and would look at why "very experienced staff failed to follow our established procedures".
"Following disciplinary hearings, two employees are no longer in our employment. The family and North Somerset Council have been kept fully informed throughout."
A North Somerset Council spokesman said Dignity had kept it "fully informed during the process".
"We are satisfied that they have investigated the matter thoroughly."

The 24-year-old has agreed a two-year-deal and will arrive following his commitments with the Stormers and Western Province in South Africa.
He is the son of former Springbok Guy Kebble and won the 2012 Under-20s World Championship with South Africa.
"I try and bring an edge to the game and make an impact," Kebble told the Pro12 club's website.
"I've watched all of the Glasgow matches in the Champions Cup this season and northern hemisphere rugby is getting very exciting. I'm looking forward to playing in a competitive European league.
"I know Dave Rennie is one of the best coaches in the world, so it's an exciting prospect to work under him next season."
Kebble will join current team-mate Huw Jones in Glasgow, with the Scotland centre signing a two-year contract with the Warriors earlier this month.
"Huw and I live together in Cape Town," he explained. "We didn't really talk about it too much before it happened, but now it's nice to know there will be a familiar face in Glasgow."
Last week, BBC Scotland revealed Kebble's expected arrival, with the new recruit considered a project player by Scottish Rugby, who have monitored him for several years.
Adam Voges and Dawid Malan both hit tons as Middlesex dominated Hampshire on day one at Merchant Taylors' School.
[NEXT_CONCEPT]
The ashes of a stranger were scattered in front of a family by two cemetery workers trying to cover up a mistake.
[NEXT_CONCEPT]
Glasgow Warriors have confirmed the signing of loose-head prop Oliver Kebble for next season.