Hampshire 27 for 2 trail Middlesex 252 (Andersson 92, Stevenson 4-71) by 225 runs
Devon-born seamer Stevenson, recalled for only his sixth first-class game, took 4 for 71, doubling his first-class wicket tally in the process, but Andersson, a man with just eight first-class matches behind him, hit 92 to take the hosts to 252 all out.
Liam Dawson, Keith Barker and James Fuller were also among the wickets, seemingly justifying Sam Northeast's decision to bowl first, but James Harris took two wickets to leave the visitors 27 for 2 in reply after a seesaw day of action.
Northeast's decision to field raised eyebrows given their hosts opted for two spinners on a pitch expected to turn late in the match.
However, in humid conditions and thanks to some early seam movement, it didn't take the visitors' attack long to reward their captain's faith. In just the third over Barker bowled Sam Robson for a duck with a beauty which clipped the top of off stump.
Nick Gubbins, fresh from his hundred in the win over Surrey also didn't stay long. The left-hander got away with one top edge hook over the slips, only to play the shot again in the same over and hole out to Felix Organ at fine leg, so giving Fuller a wicket against his former county. And when the Stevenson got in on the act for the first time, trapping home skipper Stevie Eskinazi for 18, Middlesex were struggling at 41 for 3.
That was Andersson's cue to enter the fray and he played positively from the off, steadying the ship in the company of opener Max Holden either side of the lunch interval.
Holden was dropped at slip by Ian Holland off the luckless Barker early in the afternoon, but the drop wasn't costly as the same combination accounted for the left-hander for 36 shortly afterwards - a wonderful catch taken just millimetres from the turf to end a stand of 44.
Andersson was given his one life on 23 when Dawson spilt a difficult chance at slip and he went on to make the most of his reprieve, reaching 50 from 71 balls with eight fours. He found a staunch ally in wicketkeeper John Simpson, who after a sticky start hit Barker for three successive fours.
Dawson dropped Simpson on 26, but like Holland earlier atoned soon afterwards to give Stevenson a second wicket and end a stand of 93 for the fifth wicket. Dawson, back from England ODI duty then struck with the ball having Harris taken at slip off the last ball before tea.
Six more boundaries took Anderson past his previous best of 83 against Lancashire at Old Trafford last September and to within touching distance of a maiden first-class hundred, but he came up eight short when Fuller trapped him lbw.
Tom Helm and Nathan Sowter added useful runs for the eighth wicket before Stevenson returned to dismiss the latter and Thilan Walallawita off successive balls to cap his excellent day.
Helm ensured a second batting point by striking Dawson for a towering straight six, but Dawson had him caught and bowled later in the same over.
With 14 overs left to bat, Hampshire looked set to get to the close unscathed before Harris found the edge of Organ's bat and Sowter took a sharp catch at second slip.
Barker was sent in as nightwatchman, but that plan backfired when Harris struck a second time with one that splayed the stumps, leaving the match intriguingly poised ahead of day two.