India were 84 for one in their second innings -- 60 runs ahead -- at tea on the third day.
Vijay, in an 'old-fashioned' opener's innings was 21 not out off 89 balls in two-and-a-quarter hours at the crease.
By contrast, Pujara's unbeaten 25 had come off just 52 balls.
While Vijay was becalmed, opening partner Shikhar Dhawan made a brisk 31.
But, having struck four fours, the left-hander failed to get over the top of a cut shot off Ben Stokes and was well caught by a diving Joe Root at backward point for 31 to leave India 40 for one.
It was to be the only wicket India lost in the session.
Pujara stylishly cover-drove Stokes for four, and when Liam Plunkett over-pitched, drove the fast bowler down the ground for another boundary.
It took Vijay 57 balls to both score his first four and move into double figures, with a leg-glance off Stuart Broad.
Generally, England's seamers again bowled too short and wide, allowing Vijay in particular to leave a series of harmless deliveries.
Earlier, Bhuvneshwar Kumar took six for 82 as he recorded his Test-best figures for the second time in as many matches.
But even Kumar could not prevent England gaining a first-innings lead.
England's tail wagged effectively as they added exactly 100 runs to an overnight 219 for six before the hosts were bowled out for 319.
Fast-medium bowler Kumar's return surpassed his haul of five for 82 in last week's draw at Trent Bridge -- the seamer's first Test outside of India.
- Plunkett hits out -
But nightwatchman Plunkett's Test-best 55 not out off 79 balls including helped leave England 24 runs ahead on first innings.
Together with No 11 James Anderson, fresh from his Test-best 81 at Nottingham, he added 39 in 43 balls on Saturday.
England resumed on a total owing much to Gary Ballance's 110.
Despite early morning rain play started on time, although the overcast conditions saw the floodlights at Lord's switched on.
Plunkett, who started the day four not out, began with a flurry of boundaries including a back-foot cover-drive off Ishant Sharma and one off the front foot against Kumar.
Matt Prior (23) fell when a top-edged pull off Mohammed Shami was well caught by Dhawan running back from the slip cordon.
Kumar, who overnight had taken four for 46, reduced England to 280 for nine with a burst of two wickets for four runs in three balls.
He secured his place on the Lord's honours board with by bowling Stokes for a duck with a fine away-swinger to the left-hander Kumar then had Broad edging to Dhawan at first slip.
But Anderson took England into the lead with a straight-driven four off Kumar.
However, Anderson was out for 19 when he reverse-swept left-arm spinner Ravindra Jadeja to Ajinkya Rahane at slip.
Anderson ran off quickly to prepare for his bowling stint, not glancing at Jadeja.
The two players could both be banned from matches during the rest of this series after they were charged under the International Cricket Council's code of conduct for an alleged clash at Trent Bridge.