In reply, opener Luke Wells was out cheaply for six, but Jennings and Josh Bohannon saw off the new ball and fought hard to put on 90 for the second wicket.
But, as so often, Jennings was the key wicket. When Dawson got him LBW playing back, Hampshire sensed an opening.
Teenager Rocky Flintoff went for a 10-ball duck – a second in a row after a first-baller against Surrey – as he was caught trying a big shot against Dawson before Matty Hurst was snaffled at bat pad for four.
John Turner then removed Bohannon (43) and George Balderson for seven to have Lancashire reeling at 122-6.
George Bell and India all-rounder Venkatesh Iyer (27) battled to put on 48 for the seventh wicket before a brilliant catch from James Fuller at cover saw off Iyer.
Dawson claimed the last wicket of the day, having Tom Hartley caught for two attempting a big hit down the ground.
Bell remains on 33 but, trailing by 196, Lancashire face a big fight over the next two days to take anything from this game following Dawson’s huge contribution in his 200th first-class match.