Resuming on 111-4, Voges moved his overnight score on from 29, occupying 196 deliveries to frustrate Hampshire's attack, led by Kyle Abbott (5-59).
As Voges, James Franklin and John Simpson fell to the new ball, it left them on 215-8 and Hampshire in hope of a win, with the lead at just 133.
But the Middlesex tail held on and hands were shaken on a draw after tea.
Middlesex were the only side to remain unbeaten over the entire 2016 season as they claimed their first title since 1993, but they looked in danger of beginning a new campaign with a loss when they got under way.
Without the injured Fidel Edwards, Abbott - who took his total to 16 Division One wickets in two matches - led Hampshire's attack, with Brad Wheal (3-73) also contributing.
Voges survived an lbw appeal on 29 before nightwatchman Steven Finn (16) was caught off a Wheal bouncer, but Simpson joined the former Australia Test batsman to stop any Hampshire momentum.
However, with the new ball in hand, Abbott gave the hosts a glimmer of hope, having Voges caught behind and Franklin taken in the slips before Gareth Berg snared Simpson (22) for wicket-keeper Lewis McManus' fourth catch.
It wasn't to be for Hampshire though as Toby Roland-Jones (25), Ollie Rayner (17 not out) and Tim Murtagh (19 not out) took Middlesex to a lead of 196 and safety at tea.
Michael Carberry and Jimmy Adams were brought on to bowl, signalling their intentions to agree on a draw, which brought both sides 10 points and moved Hampshire into second place in Division One behind Surrey.
Hampshire captain James Vince told BBC Radio Solent:
"The way the guys fought back from Middlesex being 220-2 in their first innings to get into a position where we were really close to being able to chase a gettable score on that wicket is an incredible effort from everyone.
"The bowlers, having lost Fidel early in the first innings, were outstanding to put in a shift to get us into that position.
"We did exactly what we wanted with the new ball and came within one wicket of probably setting up a 150-run chase in 40-odd overs. So it's huge credit to all the boys."
Middlesex head coach Richard Scott told BBC Radio London:
"We showed the old resilience of last year when we were up against it.
"When the new ball came, a couple of quick wickets fell, but Ollie Rayner, Toby Roland-Jones and Tim Murtagh were magnificent.
"We put ourselves under the pump this game. We dropped some crucial chances - if we'd have taken those chances we'd at least of had parity after the first innings. But that wasn't to be and I thought they played very well."

Christopher Michael Williams, 51, of Acrefair, died after being struck by a half-tonne power press on 3 December 2012 at Vauxhall Industrial Estate.
Mold Crown Court heard the lifting operation was unsafe and Mr Williams was not properly trained in lifting non-standard loads.
Morgan Technical Ceramics Ltd had admitted failing to ensure his safety.
The court heard a maintenance supervisor was moving the power press on a pallet truck when it toppled over, striking Mr Williams.
He died from his injuries at the scene.
As well as the fine, the firm must also pay £23,000 in costs.
Adam Voges scored 92 as Middlesex's lower order kept Hampshire's bowlers at bay to earn a draw for the champions.
[NEXT_CONCEPT]
A Wrexham company has been fined £180,000 after a man was crushed by heavy machinery at a Ruabon factory.