Right-arm seamer Dale, who is on a rookie contract with the club after coming through their Academy system, took two wickets in two balls and finished with 3 for 20 while leg-spinner Crane took the last three wickets to fall as Sussex were dismissed for 176 in 61.1 overs after winning the toss.
The hosts fought back in the final hour though, taking three late wickets as Hampshire reached 77 for four at stumps.
George Garton, who made an unbeaten 54 for Sussex, said: “It was a great fightback in the last hour and hopefully we can kick on tomorrow.
“I was just pleased to contribute. I’ve talked a lot to (head coach) Jason Gillespie about my role at no.7, I do see myself as more of an all-rounder now. It’s a different challenge but I am enjoying it.
“I thought Jack Carson bowled really well. There’s some rough created by Keith Barker and it obviously helps to get a wicket with your second ball. It is difficult to judge what a good score is. We’re coming into our first game in August instead of April but hopefully we can keep Hampshire under pressure tomorrow.”
It had been hard going for the Sussex batsmen and they were indebted to opener Phil Salt, who top-scored with 68, and George Garton, who held the lower order together with an undefeated 54. The pitch offered few terrors and there were quite a few soft dismissals.
There had been no hint that of the stranglehold Hampshire’s attack would go on to impose on the batsmen when Keith Barker and James Fuller leaked 25 runs off their opening overs but left-armer Barker then settled into a probing spell with his next seven costing just four runs.
He made the breakthrough in the seventh over when Tom Haines (14) padded up to a ball that swung a touch and trapped him lbw.
Ian Holland struck next in the 17th over. Harry Finch, who had taken 22 balls to get off the mark but was beginning to settle, played down the wrong line and was lbw for eight to leave Sussex 56-2.
Dale had initially struggled to get his line right and he was a touch fortunate with his first wicket when Tom Clark (2) mis-timed a pull to a ball outside off stump and Barker dived forward at mid-on to take an excellent low catch.
But there was nothing wrong with his next delivery which moved late off the pitch to Ben Brown, who feathered an edge to wicketkeeper Lewis McManus. Holland struck again with the last ball before lunch when Delray Rawlins was leg before for a duck, leaving Sussex 73 for five and in some strife.
Salt was typically aggressive when given the opportunity and had progressed to 68 from 83 deliveries, with 11 fours, when he toe-ended an attempted pull and Fuller took an excellent diving catch in his follow-through.
Ollie Robinson was badly dropped at third slip by Felix Organ off Barker on four but only added seven more runs when he played on to the fourth ball of Dale’s second spell to leave Sussex 125 for seven.
Sussex debutants Jack Carson and Henry Crocombe, two of nine players in the home team who have come through their Academy system, helped Garton add 31 for the eighth and ninth wickets, but both fell during an impressively controlled spell from Crane. Carson (5) fatally played back to the top-spinner and Crocombe (2) was smartly stumped by McManus after being lured forward.
Garton, though, twice hit Crane for six to loosen the shackles before Mitch Claydon (3) swung across the line to give Crane his third wicket and figures of three for 23. Garton’s excellent unbeaten 54 came off 107 balls with five fours and was his third successive first-class half-century.
Sussex fought back with the ball. Felix Organ was caught off a no ball from Ollie Robinson on 14 had not added to his score when former Kent seamer Mitch Claydon found a feint inside edge to claim his 300th first-class wicket.
Another debutant – off-spinner Carson – made an even quicker impression when Joe Weatherley (19) edged his second ball in first-class cricket before Robinson came back to bowl skipper Tom Alsop (21) and have night-watchman Crane caught behind for a duck.
Earlier, both sides had observed a minute’s silence before play started to remember the victims of the Covid19 pandemic. They then took the knee in support of the Black Lives Matter movement.