England's Bashir takes three wickets to rattle India in 4th Test
With India still 222 runs behind England's first innings of 353, Jaiswal was at the crease with Sarfaraz Khan on one at the end of the second session in Ranchi.
Joe Root hit an unbeaten 122 for England, who are fighting to stay alive in the five-match series at 2-1 behind.
Fast bowler James Anderson sent back Rohit Sharma, caught behind for two before lunch, but the left-handed Jaiswal rebuilt the innings in an 82-run stand with Shubman Gill, who made 38.
Jaiswal, who tops the series batting chart with more than 600 runs, including two match-winning double centuries in the second and third Tests, raised his bat to applause after reaching his fifty.
The 22-year-old finished the previous Test, his seventh, with 861 runs, needing 139 more to become the fourth-fastest to 1,000 runs.
Jaiswal survived a close call on 40 when he appeared to edge an Ollie Robinson delivery to Ben Foakes behind the wicket but the third umpire denied England's appeal after many replays.
Bashir, who missed the opening match due to a visa delay, sent back Gill and Rajat Patidar (17), both out lbw.
The lanky spinner then got Ravindra Jadeja caught at short-leg for 12.
Anderson moved to 697 wickets in the first session after Root ran out of partners and England were bowled out.
The 41-year-old is poised to become only the third bowler to enter the 700-wicket club after Sri Lanka's Muttiah Muralitharan (800) and the late Australian spin wizard Shane Warne (708).
England resumed on 302-7 and Root put on 102 runs with overnight partner Robinson, who made 58, but the latter's departure brought a quick end to the England innings.
Left-arm spinner Jadeja returned figures of 4-67 and wiped out the tail.