Opener Lyth, 30-years-old, reached 100 for the 23rd time in his first-class career as Yorkshire strengthened their grip on this penultimate round clash against a visiting side whose Division One safety was mathematically confirmed by Worcestershire’s defeat at Essex.
The left-hander advanced from 60 overnight to 134 not out as Yorkshire, leading by 27 on first innings, moved from 172-4 to 287-5.
Yorkshire led by 314 when the rain arrived during the lunch break, and there was no more play possible.
Heading into day four, Yorkshire still have a realistic chance of victory, as long as there is no more weather interruption.
If they win, they will also, like Hampshire, be mathematically safe from relegation.
But five points for the draw would leave them needing a maximum of just two points from next week’s final round clash with Worcester at New Road, so they would be satisfied with that outcome as well.
Lyth has endured a largely frustrating season, with only two half-centuries prior to this.
He completed a fifth-wicket stand of 133 inside 42 overs with Jonny Tattersall, who added 43.
Both men were dropped during the 33 overs bowled - Lyth twice. Substitute fielder Aneurin Donald took the gloves following a left thumb injury suffered by regular keeper Tom Alsop.
Donald, diving to his right, impaired first slip Ian Holland when Tattersall, on 16, got an edge off Kyle Abbott as he tried to leave alone.
Lyth was then dropped twice, both times by Donald off Liam Dawson’s left-arm spin.
The first came when he was on 80 - Dawson’s first ball of the morning.
The second came when he was on 94. And next ball, there was a convincing appeal for caught behind turned down by umpire Graham Lloyd.
Dawson took the only wicket to fall during the morning, Tattersall caught at short fine-leg sweeping as the score fell to 265 for five in the 73rd over.
Lyth has hit 17 fours in his 209 balls so far, and he will be joined at the crease by Tim Bresnan on four at the start of the final day.
Umpires Lloyd and Michael Gough abandoned day three just after 2.30pm with 71 overs remaining.