Campbell made 115 in his first Test ton and put on 177 runs with Hope, who hit 103, to revive West Indies after India made the visitors follow-on at Delhi’s Arun Jaitley Stadium.
West Indies slipped to 35-2 in their second innings but posted 390 all out to hand India a victory target of 121 on day four.
The left-handed Campbell and right-handed Hope steered West Indies into the lead before Justin Greaves (50 not out) and number 10 Jayden Seales (32) put on 79 runs for the last wicket.
“I think it was a great effort by the batsmen in the second innings,” Campbell told reporters.
“Batting over 100 overs, that’s a big plus for us against quality bowlers and a challenging pitch as well. We’re all hoping that it is signs of good things to come.”
West Indies, who were 248 all out in their first innings, have suffered batting flops in recent times including being bundled out for 27 — the second lowest Test total — at home against Australia earlier this year.
They suffered a 3-0 Test whitewash to Australia and face a 2-0 defeat in India after they lost the opener by an innings and 140 runs.
India finished the day on 63-1, needing another 58 runs to sweep the series, but Campbell has kept up his hopes.
“Well, as we saw the latter part of the day, the wicket was a little bit up and down,” said Campbell. “Who knows what a few early wickets might do in the morning.”
Campbell reached his ton with a six off Ravindra Jadeja but fell to the left-arm spinner lbw while attempting a reverse sweep.
“At that time I just thought a shot was on,” Campbell said in defence of his shot selection. “Just a little bit of a miscalculation on my part.”