Leicestershire 291 for 2 (Azad 122*, Hill 109*) trail Sussex 359 (Brown 133*, Haines 71, Wright 6-94) by 68 runs
On a day shortened by seven overs by bad light, Azad finished unbeaten on 122 - his first hundred since April - but Hill's 109 not out - his third of the summer - came only a week after his 113 against Derbyshire in what has become comfortably the best season of the 30-year-old's career.
The two have shared an unbroken stand of 183 to take Leicestershire to 291 for 2, trailing by 68.
Earlier, Ben Brown finished unbeaten on 133 after five and a half hours at the crease and veteran Leicestershire seamer Chris Wright took 6 for 94 as Sussex closed their innings on 359 for 9, with Fynn Hudson-Prentice, who had been injured on Sunday, unable to complete his innings.
The former Derbyshire all-rounder was struck on the left hand by former Sussex seamer Abi Sakande and his third appearance for his new county will be his last of the season after x-rays revealed a break that will require surgery.
Sussex resumed on 310 for 4 but a double-wicket maiden by Wright sparked a collapse that saw the last five Sussex wickets fall for 25 runs.
Tom Hinley's pull was plucked out of the air impressively by Ed Barnes before Jack Carson, playing away from his body, was caught behind.
Soon afterwards, Wright bowled Henry Crocombe and Sean Hunt with consecutive, almost identical deliveries that struck off stump. Joe Sarro kept out the hat-trick ball but quickly chipped Barnes to short midwicket.
Wright has 46 wickets for the season, well over twice as many as any other Leicestershire seamer. His latest haul was the fourth of five wickets or more this season and the 17th of his career.
In reply, Leicestershire lost Sam Evans first ball in the opening over as he nicked to second slip but, on a good batting surface, Azad and George Rhodes added 103 in just under 25 overs before Sussex skipper Tom Haines beat the latter's defensive push to clip the off stump with his medium pace.
Azad's first 52 runs came off 74 balls with eight boundaries but was content to bide his time thereafter, especially as Hill - batting at four with skipper Colin Ackermann on paternity leave - was eager to seize the initiative at the other end.
Hill accelerated after tea, racing to a 67-ball half-century with six fours in seven overs before slogging the left-arm spin of Delray Rawlins for consecutive sixes over deep midwicket.
Azad completed his second century of the season from 187 balls soon after picking up a 10th four but Hill was catching him fast, reaching his from 116 balls with 11 fours and two sixes.