Hampshire reached 77-2 in their second innings at the close of the second day of their Division Two clash at Chelmsford, after dismissing Essex for 254.
Graham Napier and David Masters combined to give the hosts a slight edge going into the third day, with their opponents holding an overall lead of 20 runs.
After Napier had top-scored with 74 to guide Essex to a first-innings advantage of 57 runs, Masters removed the Hampshire openers cheaply before they staged a late recovery, with Liam Dawson and George Bailey posting a 56-run stand.
Alastair Cook, with a gritty 59 spanning more than four hours, and Tim Phillips, who was left unbeaten on 40, were the other main contributors in the home side's innings.
When Cook finally departed soon after the restart, an lbw victim of paceman James Tomlinson, Essex were still 41 adrift with only two wickets remaining.
But any hopes Hampshire may have entertained of gaining a slender first-innings lead soon evaporated, with
Napier and Phillips putting on a stand of 87, the highest of the match so far.
Napier was finally undone by the second new ball, trapped lbw by Chris Wood following a superb effort that brought him nine fours in addition to a six and was achieved off 105 deliveries.
Reece Topley stayed long enough to secure Essex a second batting point before providing Briggs with his third success of the innings, leaving Phillips stranded on 40.
Masters then raced in to remove the openers in the space of three deliveries. With the last ball of his fifth over, he saw Foakes take a magnificent catch at short-leg to get rid of Michael Carberry.
And then the paceman trapped Jimmy Adams with the second delivery of his next over with the total on 21.